King Air F90 Pilot reported an erratic fuel flow indication and suspected right engine loss during an approach to landing. After landing; it was discovered the right fuel tank was empty even though 400 lbs. remaining was displayed on the fuel indicator.

Narrative

We were approx. 8 miles from ZZZ on the visual for Runway XX at 1800 feet when the fuel flow started becoming erratic and it appeared we were losing the right engine. I immediately [requested priority handling] with the Tower they instructed me I was cleared to land on Runway XX. I landed a little fast and blew a right main tire upon landing. We taxied off the Runway onto Taxiway 1 due to no thrust in the right engine we could not turn left. I called Tower and they dispatched a tug. No emergency vehicles were dispatched. The ground crew towed us in to maintenance hangar after about 15 minute wait on Taxiway. The problem was triaged by mechanics and we had no fuel in the right tank. The fuel gauges were showing more than 400 lbs in each side. We replaced the tire and topped off the aircraft and departed for ZZZ our home base. The mechanics prepared work orders and there are log entries for the return to service for our trip home. Our mechanics are currently trouble shooting the fuel system to see if it is a probe issue; gauge issue or something else.