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It cycles about every 5 seconds or so.
This sounds like a pressure leak or a check/foot valve failure and your pumps attempting to maintain pressure. Next time the pump gets up to pressure shut the power off and watch the pressure gage. If it falls back rapidly then you have either have a leak upstream from the pump or you have a faulty foot/check valve. One's on the pressure vside and the other's on the suction side. The pressure side should be easy to check. Simply close off the house shut off. If you hold pressure the chances are good you'll find the problem in the check valve located in the suction line near the pump or the foot valve in the well. Good luck. Tom