Be prepared to get wet. IMO a piece of sand or debris has gotten lodged in the check valve preventing it from seating, thus behaving as you described.
When you look at the city water connection you should see a seal with a screen. Remove that and with the pump on push in on the little tab in the middle. Water will shoot out and hopefully clear the seat. Don't forget to put the seal with the screen back. (this as I've described should be part of your winterizing process.) If this doesn't solve the issue do as others have said and plug it or replace.
As far as grey water goes a dripping faucet, even a slow drip will fill her up in short order.