When the pump is running, if I turn on the cold water faucet the check valve opens and closes rapidly making a loud knocking sound. This problem only developed after four years of trouble-free operation.

You will need to turn off the water heater first. Shut off cold water inlet valve to tank. Open the drain at bottom with hose attached to suitable drainage area, hot water will damage plants and grass so if you want to use the water let unit cool off for a few hours or drain down past the valve level. You can open the release on the temp/pressure vale and it will help vent and allow you to know the level. Take a small pipe wrench or large channel lock pliers and remove the drain line pipe connected to the valve body. Open a faucet that is the longest run and wait. The valve can be found at most home centers and are basically all the same with the BTU and pressure rating on the label. Take with you to match length and ratings. Apply Teflon tape or pipe dope on threads to help seal the fitting and tighten up till unit is secure and the opening points down for overflow pipe installation, reseal and tighten. Close off drain and refill with faucet and valve release open, when water starts to run out of valve close and continue filling until all air is bled out. Then relight water heater or turn on power. Check for leaks and test valve for operation. The TP valve should be tested annually for safety.

turn off gas
turn off cold water to heater
connect hose to spickot
place hose end outside
turn on spickot
open hot water faucets in house
wait.
when water stops, turn on cold water for 30-60 sec or so to try to stir up any sentiment in tank
turn off cold water and let drain.
close spickot
turn on cold water
let fill till water start coming out hot water faucets then turn off hot water faucets.
turn on gas and relight pilot.
finish bleeding air from hot water faucets.
set water heater thermostat

I am assuming that you are referring to a calcium or rust build up in the water heater.
If you are having this issue you will need to turn off the water to the heater. You will need a garden hose, access to a outside water soucre, cold water is fine.
Shut off the gas or power to the hot water heater before starting.
Now you will need to place the garden house on the drain fitting on the water heater, placing the end of the hose outside and lower than the drain valve. Open the valve and drain the heater, you will need to open a hot water faucet to help drain the heater. I don't suggest opening the TPR ( Temp Pressures relief )valve due to the fact that they sometimes leak or don't reseal after opening them.
Hopefully the drain valve is not clogged, you should hear and see water exiting the garden hose.
If you are not getting any water from the drain valve you are going to have a challenging task.
If no water is coming out, my recommendation is to get the required tools to remove the hot water line from the top or side or the heater, also a few towels to soak up water A bucket is helpfull also.
remove the hot water line, then you will have to use a small hose to siphon the water from the tank. Once the tank is empty you can then remove the drain valve and replace it.
once you have replaced the drain valve you should ensure that it's closed and use a Gallon of CLR. Poor this into the water heater through the hot water outlet where you removed the pipe from. Now make sure the faucets are closed and replace the hot water line. You can now turn the water back on to the water heater.
After a few minutes open a hot water faucet to let out any air until water flows out, than close the faucet.
Let the water heater set for a few hours, then drain the tank!
when the tank drains, close the drain valve again and refill the water heater. Once again open a a hot water faucet and let the air out till the water heater fills. Again let sit for a hour.
At this point open a hot water faucet at the farthest point from the heater and let the water run for 5 min. , than close it and move to the next closest faucet and repeat until all faucets have been opened and run for 5 minutes.
If you still have a slow flow in a faucet, you may need to repeat the CLR step.
I also suggest that you remove all faucet screens and clean them.
If you still have a clog than you are going to have to remove or replace a piece of water line in the house. The blockage may be hard to find but if you have access to most of the pipe or it's visable you may be able to hear the blockage with the water on. Otherwise this will result in using some water line taps and a pressure guage to find the blockage.

follow this chart troubleshooting and fix it. God bless you
est and replace a high-limit cutoff:

1. Disconnect power to the heater at the electrical panel and post a sign warning others not to turn it on.
2. Remove the upper access panel and push the reset button.
3. Replace the access panel and turn the power back on. If the water is hot, the reset was the problem. If not, you'll need to turn the power off again and reopen the access panel.
4. Use a multimeter to test the cutoff terminals for continuity. If faulty, replace with an exact replacement part.

Drain a tank:

1. Shut off power at the electrical panel and post a sign warning others not to turn it on.
2. Close the cold-water supply valve and open a hot-water faucet somewhere in the house (to speed draining).
3. Attach a garden hose to the drain valve and run it outside the house.
4. Open the drain valve and allow all water to drain out.
5. To refill the tank, close the drain valve tightly and open the cold-water supply valve. Also open a nearby hot-water faucet. When a steady stream of water flows from that faucet, the tank is full and the faucet can be closed.
6. Once the tank is full, turn the electrical power back on.

Test and replace a relief valve:

1. Lift the spring lever on the valve to fill a small cup. Check the cup for sediment.
2. If no water spurts out, or if water continues to drip after the valve is released, replace the valve.
3. Cut power at the electrical panel and post a sign warning others not to turn it on. Close the cold-water supply valve.
4. Drain a few gallons of water from the tank.
5. Unscrew and remove the discharge pipe if used.
6. Loosen the relief valve with a pipe wrench, then remove the valve by hand.
7. Apply pipe tape to the threads of the replacement valve and screw it into the tank by hand. Tighten it with a pipe wrench. Screw the discharge pipe (if any) into the valve outlet.
8. Refill the water heater and restore electrical power.

Replace a drain valve:

1. Shut off power at the electrical panel and post a sign warning others not to turn it on.
2. Close the cold-water supply valve and drain the water heater completely (see above).
3. Unscrew the drain valve with a pipe wrench.
4. Replace the drain valve with an identical unit wrapping the end with pipe tape.
5. Refill the tank and restore electricity.

follow this steps and fix it. God bless you
Drain a tank:

1. Shut off power at the electric panel and post a sign warning others not to turn it on.
2. Close the cold-water supply valve and open a hot-water faucet somewhere in the house (to speed draining).
3. Attach a garden hose to the drain valve and run it outside the house.
4. Open the drain valve and allow all water to drain out.
5. To refill the tank, close the drain valve tightly and open the cold-water supply valve. Also open a nearby hot-water faucet. When a steady stream of water flows from that faucet, the tank is full and the faucet can be closed.
6. Once the tank is full, turn the electrical power back on.

Test and replace a relief valve:

1. Lift the spring lever on the valve to fill a small cup. Check the cup for sediment.
2. If no water spurts out, or if water continues to drip after the valve is released, replace the valve.
3. Cut power at the electric panel and post a sign warning others not to turn it on. Close the cold-water supply valve.
4. Drain a few gallons of water from the tank.
5. Unscrew and remove the discharge pipe if used.
6. Loosen the relief valve with a pipe wrench, then remove the valve by hand.
7. Apply pipe tape to the threads of the replacement valve and screw it into the tank by hand. Tighten it with a pipe wrench. Screw the discharge pipe (if any) into the valve outlet.
8. Refill the water heater and restore electrical power.

1) Copy following link:
http://waterheatertimer.org/Crossover.html

2) Add comment with clear description of water heater, fuel type, and pump?
Is this a pump problem or water heater problem?
Is this a water well pump?
Are you getting good stream of cold water at tap?
Which water heater do you have?
Is the heater new?
Or older heater that stopped working.

If you need further help, I’m available over the phone at https://www.6ya.com/expert/gene_9f0ef4df2f9897e7

the tank needs flushed out ! heres how. Water heater manufacturers recommend flushing sediment from your storage type water heater periodically.
If your water heater is gas, set the gas valve to "Pilot" to prevent the burners from coming on while you are flushing it. If your heater is electric be sure to turn off the circuit breakers. With an electric water heater, if the water level drops below the heating elements and the thermostat turns the elements on, the heating elements will probably burn out quite rapidly. Connect a garden hose to the drain valve at the bottom of the tank, Close the shut off valve on the cold inlet to the water heater. open the temperature/pressure relief valve at the top of the tank by lifting the lever. Leave the valve open. Open the drain valve at the bottom of the heater allowing the water to flow out through the garden hose. when the water is clear your finished. close pressure relief valve on top of tank and turn water back on.
Open a hot water faucet in your house, and let it run until no air bubbles come out.

Turn the heater back on, and with gas units re-light the pilot light if necessary. good-day !

If the lower heating element burns out the upper element will heat the top part of the tank then shutoff. homw depot/lowes etc should carry elements. 1 turn off breaker, remove element access cover (the 2 panels on the front of the heater) test for electricity at the reset button ( uppermost device in upper panel) to make sure power is off. 2 turn off water (I usually turn off the cold water main to stop water cross feeding into hot supply line back into heater) 3 disconnect the 2 wires on the lower element, test with continuity tester. shhould be no continuity between terminals and tank, but around 10 ohms between the 2 terminals. Check the thermostat for continuity if tank is cold thermostat should be closed ie zero ohms across the 2 terminals.
If stat is bad replace stat; disconnect wires push it up out of the spring clip insert new stat reconnect. If element is bad: drain tank. connect garden hose to tank drain valve, open tank drain valve turn water main on to clear rust sediment and get water flowing. Next remove filter screen from a hot water faucet then turn on ( a hot faucet upstairs or higher than tank drain). Now turn water main off, air will now go in through hot faucet and tank will drain, takes about an hour. Use a water heater element wrench about$10 to remove element(s) make sure no bits of rubber gasket are left in element opening- dig em out with a screwdriver if needed. Insert new element with its gasket and tighten. reconnect wires DO NOT TURN ON ELECTRICITY! BE SURE TO FILL WATER HEATER WITH WATER ! Turn water main back on -air will come out of the hot faucet for several minutes. When water runs without any air sputtering, turn off faucet and replace the faucet filter screen. check element(s) for leaks. turn breaker back on.

Check power to well pump, isolation valve is open, faulty pressure switch. If only one faucet you have a plugged screen on the end of the faucet

Since the cold water works, the problem is not the pump. Look for a blockage or possibly a major leak in the hot water system. if you have a kitchen sink and bathroom sink, are both hot water outlets not working? Does the water heater have shut off valve , or is its drain still open from when you winterized it (and the water running out to the ground instead of to the faucet?