The flight instructor stated that after an uneventful local flight in the practice area, the student pilot and he returned to the airport for some landings. During the student pilot’s first landing, he flared too high and went around. The flight instructor demonstrated the subsequent landing and takeoff. He then transferred control of the airplane back to the student pilot, who flew the next traffic pattern and landing attempt. During the landing flare, the student pilot "abruptly" moved the yoke full forward. “My reaction time was not sufficient to completely correct the control inputs, which resulted in a hard nose wheel landing, followed by a pronounced bounce." The flight instructor aborted the landing, and the airplane performed as normally expected. The flight instructor then requested that the tower controllers look at the nose landing gear in flight and they reported no anomalies. After landing, the flight instructor realized that the nose wheel was flat. Upon further inspection by maintenance personnel, it was discovered that the engine firewall was wrinkled and required major repair.