Agree with everything everyone says here, but I had a slightly different issue somewhat like this with our RV.
The water pump was cycling, frequently, about once every 5 minutes. I too expected there was a leak, or the pump was bad. Checked in detail for a leak didn't find any, so figured the pump was the cause. I was hesitant to take the pump apart, cause sometimes those things just don't take well to repairs, and since I figured I needed a new pump, I took out the old one, and took it apart, and found a small piece of silicon rubber stuck in the check valve, letting a small amount of water to leak past, causing the problem. Put it back together, and it's been working fine for the past 2 years.
I really expected it to be something like a damaged check valve.
Worst part of this, for some reason, know only to the RV gremlins, this happened at 2AM, with the pump cycling repeatedly, keeping us awake. Did turn it off then, but why start in the middle of the night?