Pilot flying M20C aircraft reported engine power loss due to low fuel situation inflight.

Narrative

Before leaving ZZZ1 I checked the tanks in the Mooney and verified that the fuel gauges were indicating what was observed in the tanks. I estimated that I had half in the left tank and over half in the right tank. The gauges seemed to be corresponding with that amount in the tanks at that time. The flight was going to be roughly 2 hours and 15 minutes to return to ZZZ2.During the flight I had been switching tanks every hour. About 20 miles south of ZZZ3 I switched to the right tank and estimated that I probably had about an hour left. Noticing that I had a little bit more of a headwind than was predicted I did another fuel calculation based on my ground speed via what was indicated on the Garmin. Realizing it might take me a little longer to get to ZZZ2 I decided at that time to divert to ZZZ3 to get more fuel before continuing on to ZZZ2. I notified ATC of my intentions and began heading straight to ZZZ3.About 6 miles south of ZZZ3 during my descent for landing on Runway XX my engine began to sputter then died. I turned on the fuel pump and tried switching tanks. I glanced at the fuel gauges; and it appeared that the right tank had more fuel and with my previous calculations determined that it was the right tank that should have more fuel at this time than the left. I switched back to the right tank and did not get an immediate response; that's when I [advised] ZZZ Center. Checking my engine indications everything seemed to be looking normal while the prop was still windmilling; but my fuel pressure had dropped to 0. During that time the engine started again which provided enough time and altitude to continue the landing at ZZZ3.Upon landing at ZZZ3 I discovered that the fuel gauge on the right side had become stuck and was not indicating accurately. What I learned is to never rely on the fuel gauges for accurate indication as they can always become inoperative. I should have assumed that it would take me longer to return to ZZZ2 and should have ordered extra fuel as a buffer and from now on will do so. The Mechanic of the aircraft was notified and since then has fixed the fuel gauges along with adding a fuel dipstick stick with that aircraft.