I've also had this "burping" issue for over two years, now. At first it was about once an hour; eventually worked its way up to ever 2-3 minutes. about two weeks ago, my water heater (on electric) was running almost constantly and making a lot of noise. I figured it had some sludge in it, so turned it off, let it cool down, turned off the pump and drained it. A lot of gunk came out
! Turned the pump back on and refilled the tank; then turned on the water heater. What a difference! Almost silent, now. It also cut down on the frequency of the "burping"/ghost-cycling of the water pump, but it still did it about once every 1/2 hour.
A couple of days later, it occurred to me that I could use one of the two low-point drains to fill the dog's outside water dish - which sits directly underneath the drain. (Don't know why it took so long to figure that out. Must be a "Senior Issue".) Anyway, Within a couple of hours, I noticed the burping was gone! Still did it a couple of times a day, so I drained a little out of the second low-point drain. Now the burping is completely gone! Don't know why this combination of actions worked, but it did! You might give it a try.