My son bought an older house with a well, we have no experience with wells. This one pumps up to 40 lbs shuts off, quickly drops to 20 lbs and kicks on again. I shut the water off to the house, and the same thing happened.
Sounds like you may have a bad pressure tank. Try turning off the well pump open a faucet in the house and let all the pressure out. When the water stops running take an air pressure gauge and check the pressure in the tank. It should be 2# lower than what the pressure is when the switch turns on the pump.
You stated that the switch is cycling 20 / 40 so your tanks air pressure should be set at 18#.
If the air pressure is lower you can add air, If you get water out the schrader valve you need to replace the tank.
Sometimes you may do this a couple times only to find the tank is bad.
Replaced pressure tank today, set pressure at 18. Pumps up to 40 and starts to drop right away. Can you put a check valve in between the well and the tank to hold pressure. Thanks
Why do you ask, do you know the water is going back down the well? If so, you should address that problem. You may have a hole/crack in your droppipe, a bad check valve or a broken supply pipe from well to house.