With the valve to the house shut off, my pump comes on at 20psi, takes a
minute to pump up to 42psi, shuts off, and takes 4 minutes to work down to
20psi, ad infinitum.

I have a large diameter pipe that sticks out of the ground about a food. At
ground level inside the pipe is a metal box that says Square D. About two
feet below that a white plastic box is attached to the side of the pipe;
presumably the pipe to the house comes out of the plastic box. A stream of
water is coming out of the side of the box.

The cap to the well says I have a Franklin submersible 2 wire 120v 7a pump.

I normally do all my own plumbing (I've installed water heaters and pressure
reduction valves....), but obviously don't know anything about wells.
Presumably the plastic box is cracked and has to be replaced, but I am not
even sure how to get at it.

Any help, or even links to good websites, would be appreciated.

In the mean while, we only have the pump on about 3 days a week, and it only
runs 25% of the time, so that is like leaving a 100w bulb on. That's not so
bad, is it? Reason I mention this is that the pump was old when we bought
the house 20 years ago. If it is not something I can fix, it might be cost
effective to just let it run until the pump dies and then have a plumber fix
everything at once, rather than coming out twice.