Starts fine when cold, but after parking and at certain temperatures why do I have to give it gas to get it to catch?

Check the ECM grounds, they are under the passenger side kick panel. When the engine goes to warm status, the Oxygen sensors come on, If there is no ground on the ECM then it starts pulling ground voltage across the sensors and that causes big issues.

No codes will show but one of the mechanics would have known whats wrong so try another dealer ,i suspect a temperature sensor is giving the wrong information to the computer but as i have never seen one of these vehicles i couldnt tell you what and where sorry about negative answer but ?? now a citreon or a renault i can answer

Have you checked base fuel pressure with a gauge yet ?
Sounds like the pump is weak.

Wrong fuel octane water in fuel...

The sender the Expert is referring to is a temperature sensor that tells the computer your engine is cold. The control then enrichens the fuelmix, sets timing and goes into a run cycle after the engine is warmed up.

You instead may have a leaky fuel injector. This is bad because the gas that floods the engine can go into the oil pan and dilute the oil which eventually ruins the engine. A fuel line pressure reading is taken when the car is parked and then the fuel pressure is read later that day or the next morning.

If the pressure drops there is a problem. Some shops have machines which can benchtest the injectors when they are taken out. A single injector can be the whole cause or all may need cleaning or replacing.

There is a chemical that Dealers or Repair shops can insert into the fuel system by disconnecting your fuel source and inserting this can of chemicals. The engine is operated for a few minutes using only the cleaner as fuel. The car is suppose to sit for several hours or overnight for the chemical to finish working. This may be all you need if it works.

The reason I have made my comment is that you are getting a rich mix at startup. If all starts warm and cold are causing trouble it can be either my solution or the other Experts. The engine needs a warm signal to end the "cold start cycle" for later starts. But the difference is that the initial "cold" start ought to be fine if the "cold start" cycle was stuck ON. Leaking injectors would flood the engine between starts and on the initial "cold" start making my solution an accurate assessment of your problem.

I hope you find my idea useful and would like to know the outcome of your problem.

I live in the snow belt and have had the same problem for two years. Car runs and starts fine in the summer but starts giving problems when it gets cold and damp. After having it diagnosed with no problems indicated, then having the plug wires, plugs, etc. replaced, I discovered that if I put the key in the ignition, turn it two clicks and leave it for several minutes it will start and run fine. When it's just cool and damp, it may only take 2 mins. but the colder the weather the longer it takes. At 9 degs. it took about 20 mins. Hope this helps. It's a very expensive and frustrating problem that no one seems to know anything about.

Pumps either work or they don't. yes this car has one pump (sending unit in the tank) main pump I believe its under the car.
One more test before I'll call it at the cold start system. Remember these tests are for a gas operated vehicle NOT DIESEL.Take the car for a short spin enough for it to get to operating temperatures then park it at an inside garage or somewhere where the temperature is above 60F.
leave it there for an hour or two then start the car. If it starts w/o a problem its either the thermo time switch or some other control in the cold start system. That switch tells the computer to give the vehicle more gas because its below a certain temperature.

these cars are carburated motors and need around 10 minutes to warm up because they operate at higher temperatures.

could be trash in fuel filter or around pickup screen in tank. While driving everything sticks to them like magnet when you shut off and it sits it all falls off to happen again when you start the vehicle again