It sounds like either the city water is running backward through the pump because its internal check valve is stuck open a bit, or it could be the Anderson Valve leaking.
Assuming you have one, you can check the Anderson Valve by disconnecting the line hooked to the right-hand side (looking from the inside) of the fitting to see if water is coming out. To do that, you'll have to take down the rear basement wall near the Universal Docking Center (UDC).
I've attached a picture of the backside of an Anderson Valve.
If water's coming out the connection on the right, to the Fresh Tank, the valve assembly, or valve cartridge needs to be replaced. Not hard to do and Heartland or Anderson will ship you the parts for self-install.
If there's no water coming out, or if you don't have an Anderson Valve, the water pump is the culprit. Also an easy self install and it's behind that same wall. Alternatively, you can pick up a 1/2" press-on Sharkbite check valve and add it on the suction side of the pump. You can find one at many hardware stores.
Until you get it fixed, it's probably a good idea to shut off city water and run from the fresh tank using the pump.