My coolant temperature sensor (Cylinder Heat Temperature Sensor) keeps popping out of place.

The Engine Coolant Temperature Sensor is located in the coolant pipe, below the air intake pipe and behind the right cylinder head and is integral with the ECT sensor. The Engine Coolant Temperature Sensor is located on the back of the right cylinder head and is integral with the ECT sensor.
https://www.autozone.com/repairguides/Audi-Cars-1999-05/Component-Locations/Engine-Coolant-Temperature-ECT-Sensor/_/P-0900c152801c0f85

What engine ? 3.6 or 4.6
Engine Coolant Temperature (ECT) Sensor (LH2)
In the rear side of the right cylinder head
Engine Coolant Temperature (ECT) Sensor (LY7)
In the left cylinder head near the center cylinder

Temperature Warning Light is on Inspection Service How this system works: With the exception of older, air-cooled vehicles, cars rely on a liquid called coolant (or antifreeze) to keep the engine at an optimal temperature. Coolant is a mixture of water and ethylene glycol, and it circulates around the engine block and absorbs excess heat, which keeps the engine from overheating. The coolant converts that heat to air in the radiator, and then the air is emitted, at which time the coolant is ready to absorb more heat. Without coolant, the engine would quickly ruin itself with its own heat production. Common reasons for this to happen:

• Coolant is low or weak: The most common culprit when an engine overheats is low or weak coolant. If your car doesn't have enough coolant, then it can't absorb enough heat. While your car will naturally lose small amounts of coolant over the years, a leak is the most likely cause of low coolant levels.The proper ratio of coolant to water can also get distorted, resulting in a problem. Too little or even too much antifreeze can dramatically lower the boiling point of the coolant. A proper ratio of antifreeze to water is 50/50 to 60/40, depending on the vehicle.
• Broken radiator fan shroud: The radiator fan shroud directs the airflow across the radiator so the air can absorb the coolant's heat. When the fan shroud breaks or becomes dislodged, air fails to enter the radiator, and the coolant will no longer have a place to direct the transfer of heat.
• Broken or missing air dam: Along with the shroud, some vehicles have an air dam (or deflector) underneath the vehicle. If this is broken or missing then the air can pass underneath the vehicle but not also through the radiator, which will cause overheating. These air dams are essential in newer vehicles, as they force the air through the fan shroud.
• Faulty coolant temperature sensor: The temperature sensor takes constant readings of the coolant temperature and sends that information to the engine control unit. Based on the temperature of the coolant, the engine control unit adjusts the ignition timing, the fuel injector pulse, and the operation of the electric cooling fan.
• Bad water pump: The water pump is responsible for keeping the coolant cycling throughout the engine. After the coolant transfers its heat energy to the air, the water pump recirculates it around the engine so that it can absorb more heat. The most common water pump problems are a leaking pump, bad bearings, or an impeller that has rotted away due to a low coolant ratio.
• Stuck thermostat: The thermostat acts as a dam for the coolant. When the engine first turns on, and it is still cold, the thermostat keeps the coolant from circulating, which allows the engine to warm up as quickly as possible. Once the engine has reached its operating temperature, the thermostat opens and allows the coolant to circulate. A stuck thermometer may stay permanently sealed and therefore keep the coolant from reaching the engine block.
The thermostat may also stick open. This will not usually result in overheating, but it will waste gas.

• Broken engine cooling fan: The engine has a cooling fan that is deployed when the coolant needs some extra help. When the coolant temperature sensor notices that the coolant temperature is getting too high, the engine control unit (on newer vehicles) will initiate the cooling fan to reduce the temperature.
• Broken thermostatic fan clutch: Older vehicles use a thermostatic fan clutch to engage the engine cooling fan, which is mounted to the fan blades. The fan clutch uses a bi-metallic spring that tightens when the temperature increases. This acts as a "high speed" option for the fan, and when engaged, it draws more air across the radiator.
• Blown head gasket: The head gaskets sit between the engine block and the cylinder heads, and keep coolant from entering the engine's oil and combustion chamber. When a gasket blows and coolant seeps in, the issue is not only that the engine will overheat, but also that damage may be done to the catalytic converter and oxygen sensors due to contamination from the coolant. What to expect: A top-rated mobile mechanic will come to your home or office to determine the cause of the temperature warning light turning on and the source of the overheating, and will then provide a detailed inspection report that includes the scope and cost of the necessary repairs. How important is this service? An overheating engine is extremely dangerous. It is not safe to drive a vehicle with an overheating engine, or you may ruin the engine completely and put yourself at risk. As soon as you notice the light come on, pull over. If there is no place to safely pull over, turn off your radio and other electrical units, and turn your heat on high (this will funnel some of the hot engine air into the cabin). As soon as you can safely pull over, do so, and then book one of our mechanics to perform an inspection.
  

Does it run when the engine is cold, how about with just the key on engine off

The code says that engine temperature was not reached to allow closed loop operation. Basically the coolant temperature sensor did not inform the ECU that the engine had reached normal running temperature. 4 things to check: 1) coolant level 2) coolant sensor cable and connection 3) coolant sensor 4) thermostat The first two are fairly easy. To check the coolant sensor (3) drain the coolant into a bucket (you can reuse it when you refill). Put a voltmeter across the connector pins on the sensor. Now immerse the end of the sensor in a hot cup of water and again measure the voltage output. You are looking for a change either up of down in response to temperature change. If you see a change the sensor is OK, if not change it for a new one. Whilst the coolant is drained take out the thermostat and drop it into a pan of simmering water. The 'stat should pop open. Take the pan off the heat and after about 5 minutes the 'stat should pop closed, if it does it is OK. If the 'stat remains open then it is faulty and this is the source of the problem in that coolant circulation is unimpeded and the engine is is being kept below operating temperature.

Replace the Engine Coolant Temperature (ECT) sensor. ECT sensor is placed in the right back side of the cylinder head on 2002 VW Jetta 1.9 TDI. See picture below:



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DTC P0128: Engine Coolant Temperature (ECT) Below Thermostat Regulating Temperature
Do you have any issues with heat in the vehicle? If so the thermostat may need replaced If not the engine coolant temperature sensor is most likely reading incorrectly due to a bad sensor or connection to the sensor.

According to Autozone.com:

Because ECT (engine coolant temperature) sensor procedures vary depending upon their location, first locate the part. You may need to partially drain the engine coolant to prevent coolant spillage.

• Disconnect the negative battery cable.
• Unplug the ECT.
• Then, unscrew it.
• Reverse the removal procedures to install the new part.
• Connect the negative battery cable to the battery.

Related Parts:



Cooling Fan Switch
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Engine Temperature Sensor
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Temperature Switch
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Coolant Temperature Sensor

Fig. Remove the coolant temperature sensor
The coolant temperature gauge sensor is a temperature-variable resistor, or thermistor. As coolant temperature increases, the resistance of the sensor decreases or decreases, depending on the type of sensor.
A1 and A2 platforms use a different type of circuit that A3 vehicles. On A1 and A2 vehicles, the circuit is a "resistance to ground" type. A3 vehicles use a "variable voltage" type, where a voltage is supplied to the sensor. Because of the circuitry design on A3 vehicles, testing of the coolant temperature gauge is limited.
The engine coolant temperature gauge uses a heat sensitive sending unit to transmit an electrical signal to the gauge. The sending unit is a heat sensitive variable resistor that is located on or near to the cylinder head and threads into an engine coolant passage. The sensors are a Negative Temperature Coefficient (NTC) type. As the temperature increases, the electrical resistance of the sensor decreases. As the coolant temperature changes, so does the resistance of the sensor. The gauge is calibrated within the operating range of the sensor and interprets the resistance value to display the coolant temperature.
Beginning with model year 1994, the engine coolant gauge and the Engine Control Module (ECM) temperature sensors were combined into one sensor with 4 terminals. The basic operation remains the same in that their resistance decreases as the coolant temperature increases, however the actual resistance values of the 2 sensor circuits are different. The electrical connector of the 4-wire terminal sensor ( 1 and 2 ) is keyed to prevent improper connection of the sensor's electrical circuit.

Fig. Beginning with model year 1994, the Engine Coolant Temperature Sensors combined the sensors for the ECM and the gauge into one sensor-1997 2.8L V6 sensor shown

Fig. The electrical connector for the combined temperature sensors is keyed to avoid improperly connecting the sensor's wiring-1997 2.8L V6 connector shown

4.2L Engine To Remove:

1. Disconnect the negative battery cable from the battery.
2. Drain the coolant from the engine.
3. Disconnect the engine coolant temperature (ECT) sensor connector. Engine coolant temperature sensor removal and installation
   
   
4. Remove the engine coolant temperature (ECT) sensor from the engine.

To Install:

1. Apply thread sealer P/N 12346004 or equivalent to the engine coolant temperature (ECT) sensor threads.
2. Install the engine coolant temperature (ECT) sensor into the engine. Torque the sensor to 12 ft-lb (16 Nm).
3. Connect the engine coolant temperature (ECT) sensor connector.
4. Refill the engine with coolant.
5. Connect the negative battery cable to the battery.

5.3L & 6.0L Engines To Remove:

1. Drain the coolant from the engine.
2. Disconnect the engine coolant temperature (ECT) sensor connector. Engine coolant temperature sensor removal and installation
   
   
3. Remove the engine coolant temperature (ECT) sensor from the left cylinder head.

To Install:

1. Apply thread sealer P/N 12346004 or equivalent to the engine coolant temperature (ECT) sensor threads.
2. Install the engine coolant temperature (ECT) sensor into the left cylinder head. Torque the sensor to 15 ft-lb (20 Nm).
3. Connect the engine coolant temperature (ECT) sensor connector.
4. Refill the engine with coolant.

Coolant Temperature Sensor/Switch (For Computer): Service and Repair
ENGINE TEMPERATURE SENSOR
Removal




Engine temperature sensor, 5-cylinder
Jack up the car.
Remove protection cover from under engine.
Drain the coolant.
Lower the car.
Release coolant hose from thermostat housing using pliers.
Remove thermostat housing.
Remove sensor.




Engine temperature sensor, 6-cylinder
Jack up the car.
Remove protection cover from under engine.
Drain the coolant.
Lower the car.
Loosen oil reservoir for servo pump.
Remove coolant reservoir and put it to one side.
Remove screws and outer/upper timing-gear casing.
Remove screws and wire channel.
Disconnect connector.
Remove 4 screws and thermostat housing.
Remove sensor and wires.
Installation
Engine temperature sensor, 6-cylinder
Install thermostat housing sensor.
Position wires in slot on thermostat housing.
Note! The white marks should be positioned in lower opening of slot, from underneath.
Position gasket on thermostat housing correctly and install thermostat housing using two of the screws.
Position thermostat housing correctly and tighten the four screws.
Position wire channel correctly and position wire in channel.
Tighten screw on wire channel.
Connect the connector.
Install outer/upper timing-gear casing.
Install coolant reservoir.
Install oil reservoir for servo pump.
Follow-on work, 6-cylinder
Fill up coolant.
Warm the engine until the thermostat opens.
Switch engine oft and check the level. Re-fill if necessary.
Check for leaks.
Check in VADIS vehicle communication (read-out of parameter values), for current engine system, that the engine temperature shown appears
correct.
Engine temperature sensor, 5-cylinder
Install in reverse order.
Follow-on work, 5-cylinder
Fill up coolant.
Warm the engine until the thermostat opens.
Switch engine off and check the level. Re-fill if necessary.
Check for leaks.
Check in VADIS vehicle communication (read-out of parameter values), for current engine system, that the engine temperature shown appears
correct.