The water pump check valve is INTERNAL.....diaphragm type
The city water check valve is inside the city water connection...plunger/spring/o-ring type (when they leak water flows out of connection when using pump....air does not get sucked back in)
Cap off that wintering valve.....
Check pump inlet strainer.........tight not sucking air
Check pump suction fitting.......not cracked sucking in air
Check connections on that winterizing hose/valve.tight not sucking in air
With water in fresh water tank turn pump on.open closet faucet hot side so air can be expelled and water heater fill up...then cold side.