PA-38 pilot reported a loss of lateral control on the ground during a touch and go due to a flat nose wheel tire. Pilot regained control but incurred a prop strike when exiting the runway.

Narrative

I was in the Pattern for a 3 lap currency event; on the second landing I had an impressively smooth touchdown upon the numbers. As I began to power up and initiate the touch and go I felt a sudden sharp braking and flutter of the noise wheel. I initially tapped the brakes thinking they were frozen or locked up; I began to lose control of the aircraft making S turns down the runway; and I powered down to idle. I immediately notified ZZZ Tower of the situation and I tried to roll the aircraft with a flat nose wheel into a taxiway and OFF the runway; in the process there was a minor prop strike. I cooperated and complied with both them and the ground workers. The aircraft was towed back to the flight school. I was uninjured.