RV Basic Water 101
'City water' and on-board pump discharge feed the SAME cold water line....cold water feeds to toilets and sink/shower plus inlet to Water heater 't' off that main cold water line.
Water heater HOT out then feeds hot water to sink/shower.
City water connection has valve to switch over to 'FILL' position. That will direct city water to fresh water tank for filling.
Normal position will be on 'city water' (FILL only when fresh water tank needs water)
City water inlet has a check valve to stop water from backflowing out when using on-board pump.
On-board pump has an internal check valve to stop backflow thru pump when using city water.
Fresh water tank can fill/overflow if pump check valve fails.
On-board pump has a pressure switch that starts/stops pump.
If FILL valve is open and pump on ---then pump will suck water from fresh water tank and pump it thru FILL valve back to fresh water tank...RoundRobin
When on city water........
If you turn pump ON it should NOT run UNLESS city water pressure is below pump start set point.
City water inlet has a screen washer ...gets dirty and limits flow/pressure .
IF check valve fails........could prevent water flow. If using pump then water will run out of city water connection
Pump is 'on-demand'....so if a faucet/shower etc is NOT open then no demand and pump S/B OFF.
If it cycles then there is a leak.