As others have said, there should be no air in the pressurized portion of the water system. So, you definitely have a water leak somewhere and it could be anywhere from the pump forward. For the symptoms that you describe, it will be a very tiny (pinhole) leak, just a drop or two per minute.
Places to look:
- low point valves
- under the kitchen sink
- under the bathroom sink
- toilet
- shower head and mixer valve
- water heater, inside and drainplug outside
- water pump connection on the pressure side
- City Water connector, inside and outside
- bypass valve connections
- inside a wall
Things to try:
- When checking connections, use a paper towel to detect wet surfaces.
- Look for signs of water damages on floors, walls, ceilings, under the bed, and under the frame.
- Sniff for mold. If it has been leaking for a while, mold is likely. Remember, this leak will occur even when on City Water.
- Put the water heater into bypass mode and retest. If that works, the issue is with the water heater or the supply to the water heater.
- Close the hot water output valve, but leave the cold open, and retest. If that works, the problem is on the hot side of the system and you can remove the toilet from suspicion.
- Install a new valve on the cold water pipe AFTER the bypass valves, so the hot pipes are pressurized, but the cold pipes are not, then retest. If that works, you know your leak is on the cold side.
If all else fails, the check-valve inside the water pump may be malfunctioning, allowing a small backflow. You could disassembe the pump and clean it, or you could just get a new pump. This is the one in my 23H: