Q. My irrigation pump runs fine when the system is operating, but after it turns off it cycles on for 5 seconds every 10 minutes or so.
A. If you are using a pressure switch and pressure tank to turn the pump on and off my first guess would be that you have a water leak in your irrigation system. The water leaks out, which cause the water pressure to drop, then the pump kicks on and recharges the pressure. Then the pump shuts off again. That would cause exactly this symptom.
Knowing the problem is the easy part. Finding the leak, that could be harder to do. It could be a zone valve that isn’t turning off all the way or it could be a leaky pipe. You can narrow the search area a little, the leak will be someplace in the pressurized part of the system, that is, between the pump and the zone control valves. Start by looking for obvious dripping, then look for someplace that seems wetter than it should. If it is a leaky zone valve then the water will be leaking through the valve into the sprinkler zone pipes and will dribble out at the lowest sprinkler head. So look at the sprinkler heads. There will be a small “swampy” area around the lowest sprinkler head that is controlled by that valve.