C172 pilot reported a jammed stabilizer/elevator while attempting to land the aircraft; resulting in a balked landing; a blown tire; and a runway excursion.

Narrative

In the traffic pattern at ZZZ practicing short approaches. When I realized we were too low; I decided to convert the power-off 180 into a normal landing; by stopping the descent rate and adding power to get back onto glide path. We were coming over the threshold; I added power to get back onto glide path; then took power out once we were established in the flare. We were in a very nose pitch high attitude; and I was struggling to push the nose down; then the CFI took over controls; added full power to initiate a go around and wasn't able to push the elevator down either. He went power idle; and was able to force the elevator down; we safely landed on the runway. But; we only had less than 1;000 ft of runway left; so we went off the end of the runway; maintaining centerline; into the gravel. After we shut the plane down and exited safely; we noticed a piece of the stabilizer was lodged in between the stabilizer and the elevator; essentially locking it into place. The right main gear tire was also flat; possibly popped during the landing. There was no major structural damage to the aircraft; from what we examined. Neither one of us; student and instructor; sustained any injuries.