Your water pump check valve is OK....it is internal part of pump
Your city water inlet connection check valve has failed.....stuck open/o-ring dislodged etc
Install a garden hose shutoff valve on inlet connection and then connect hose to it.
Pump On...valve closed
Pump off---using city water....valve open
Simple
City water inlet check valve is where inlet hose connects on outlet side of hose reel so can be a real PITA to R/R
Garden hose shutoff valve simplest solution