A Flight Instructor reported the student lost control of the aircraft while landing resulting in a runway excursion; horizontal stabilizer damage and a flat tire.

Narrative

Student practicing power off 180 short approaches on runway XXL at ZZZ. Just as student was flaring and about to touch down and large gust pushed the aircraft right of centerline and downward towards the ground. I; as the instructor took controls and initiated a go around. The aircraft began to climb and noticed we had an excessively nose up attitude. After adjusting elevator trim I discovered I was unable to pitch down. After more force was applied I was able to pitch back for the runway and made the decision to land. With approximately 1000' of runway left. I applied heavy braking to come to a stop. The aircraft was unable to stop on the runway and carried on to the runoff area. No injuries were sustained. The aircraft's right main tire blew under braking. After inspecting the aircraft we discovered the outside edge of the instructor side horizontal stabilizer took damage. This impeded the elevator from moving causing the nose up attitude and need for forceful control to reverse. No other damage to the aircraft noticed.