New well pressure tank short cycling
Is the pressure gauge close to the pressure switch or some distance away.  Maybe the pressure switch is seeing a different pressure than the gauge. Too small piping could also contribute as well as any blockages.   Either that or the pressure switch isn't adjusted properly.   

With that small a tank the usage by the sprinkler system and the production rate of the pump would need to be very closely matched or you'd short cycling as you describe.  It sounds more like an expansion tank. 

Short cycling is also very hard on the pump motor too. 

You mentioned you recently replaced some items, was it working before? What is the reason for the changes?   

I'd put in a much larger storage/pressure tank.  That would minimize the short cycling. A diagram of your plumbing would help us sort this this out as well. 

Sorry for posting this as an "answer", but it was too long for the comment section.