Water pump shuts off for several minutes then restarts??

To my understanding, there is a pressure switch incorporated in the pump head. When it senses a pressure drop, it activates the pump. When it reaches the preset pressure, it shuts the pump down. It sounds as if it may malfunctioning. Depending upon what make/model of pump you have, it may be a replaceable component. On the other hand, depending upon how much you use the onboard pump, it may just be more prudent to change out the pump for a new one. There are a few really good higher flow models now on the market.