I do not know where the pipe with the black valve goes to or where is it coming from. I closed it for a few minutes but I was still getting water to the faucet that is after the this valve.
You are only saying this
NOW? How many times did I write about turning off the water to the home with a valve as a test, and you never said you did not do that. The purpose of closing off the water to the home was to be sure the water was not going to a big leak into a drain to the septic or something. You say that stem on valve
It seems unlikely that a massive leak like that would not be noticed, but it should have been checked for. For now, turn off the valve with the black (or blue, which it looks to me) handle. I labeled it A in the attached picture. Off would be with the handle at right angles to the pipe. Leave it off for now.
See the marked up picture I attached. I am thinking that B is the valve that should shut off the water going to the home and that valve is defective. It should be replaced with a working valve. Is it possible that the valve B works but does not have a stop? That is, when you rotate the stem, that there may be a position that stops water to the home. Maybe have a helper monitor a running faucet and shout if the water goes off while you rotate.
In the marked up picture below, I marked a pipe "From well", which I think is the pipe you are saying is coming from the well. If that is the water from the well, and if you have eliminated a big leak going elsewhere, that is the pipe the check valve, to work around a down hole leak, would go. The arrow on the check valve must point toward the pressure switch and point away from the well.
There is not a certainty that a check valve will stop this cycling. Your installation has flaws. You should have a drain valve that lets you drain sediment from the base of the pressure tank. Your pipe into the pressure tank could be clogged up. So while the check valve will probably stop the cycling for now, it may not.
I presume you are leaving the power to the pump turned off most of the time.
I am not a pro. Click Inbox above.