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The pump cycles when there is a leak. That may be internal to the pump due to debris in the diaphragms and is usually from plastic chips out of the tank. These can be cleaned out with compressed air once you take the pump out and take the head off. Not hard. It can also be from a crack in the water filter (if you have one). Naturally, you don't have the pump on when you are hooked up to city water as there is no need for that and it can damage the pump. Hope that proves helpful here. Please get back to me if we need to dig deeper into this as all follow ups are included in our fee.
Just checking back in to see if you were able to resolve this issue or if you need more input from me here?
If the pump recycles, then it is the problem if you have no leaks. The pressure switch may be defective or dirty. IF you don't have a filter on the inlet side of the pump, I can just about guarantee you there is some debris inside it. I see this question frequently here and in my own shop about once a month. You can exchange out the pump if you don't want to replace it. Not hard and a new one is around $100 or so depending on which one you get.
If you ask for one it will be provided. I think I gave you what was needed to repair it, but that may not have been something you wanted to tackle. Not a problem as RV repairs aren't that easy.