This is an extremely common problem. As noted, its a leaking check valve in the pump. You hook to city water and your fresh water tank slowly fills and overflows.
Remove the city water altogether and then run the pump extensively by opening faucets. Shut the pump off, and reconnect your city water. It probably will be ok after that. If you accidentally turn on the pump with city water on the trailer you do stand a chance of unseating the check valve and it leaking.
Zillions of Youtube videos about this. Here is one that covers it pretty good.
https://youtu.be/B_yI_tDcOxYIn fact, these two sisters, having been teachers all their lives, do a very good job with the videos, showing and explaining stuff.
Charles