I just checked out the pressure tank. It was full of water and there was no air released when I pressed down on the stem for putting air in the tank. I tried using a bike tire pump to add air but it wouldn't take any air. Also, I might have described the problem a bit wrong in the OP. When I turn on the water, the pump whines......for 10-15 seconds then kicks in. However, not all the time. As I mentioned in the OP, I've had to loosen the large cap on top (water supply intake), let some water run out, then re-tighten the cap. Maybe it is time for a replacement pump.