It makes a big difference if the pump was cycling on and off while irrigating or if it was running continuously while irrigating. If it was cycling on and off, I suspect the second check valve and leak back down the well is the reason you were getting air in the lines. If the pump was running continuously, I suspect you were pumping more than the well produces, and you were picking up air that way.
Either way the second check valve is a problem, and even more so because you have a leak down hole. With a check valve up top, it is holding water in the pipe like holding your finger over a straw full of water. Then when the pump comes on the pressure hits the above ground check valve like a boxer punching you in the nose. This creates a water hammer on pump start that you may hear as a thump or pop. The only way to solve this problem is to fix the check valve on the pump. This may also be the reason for air in the lines so you can kill two birds with the same stone, so to speak.