Water mystery on pump mode

The city water fitting is equipped with a check valve that's purpose is to stop the back flow from the pump when the city water supply is not connected. Your check valve has failed in some way. Fixing it may be as easy as easy as removing the filter screen if it has one, and pushing in on the check valve and letting it re-seat itself. If that fails, the small rubber O-ring that forms the seat may have been displaced. If you can see it and get it back in place, you'll be all set. Otherwise, you may need a new fitting. A quick interim fix is to replace your existing push on cap with a screw in plug from a yard & garden shop. An inline garden hose shutoff valve also works well.