Fresh Water System Leakage

If capping the "city water inlet" stops the pump from cycling, then the check valve in the inlet is bad. Replacements of the inlet and valve unit is readily available. You remove the inlet from the outside, but the hose may be difficult to get to depending on the slack they gave you.

The check valve is inside the fresh water connection. I believe it is about $15 or so.

These posts are common in the spring because people damage the check valve when winterizing.

A few years ago, we had the pump suddenly start cycling on its on in the middle of a 3 week trip. I tracked it down to the shore water inlet check valve letting water out. At least all of the leaked water was OUTSIDE of the RV. I bought a garden hose shut off valve and screwed it to the shore water inlet and closed the valve. That solved the problem until I could get home. When at home, I replaced the check valve.