The student pilot was landing at his destination airport. He reported that when the airplane was over the runway numbers, he reduced engine power to idle to prepare for landing. During touchdown, the airplane landed hard and bounced. The airplane bounced two additional times before coming to rest on the runway with a blown nose landing gear tire. A postaccident examination of the airplane revealed substantial damage to the engine firewall. The student pilot reported no mechanical malfunctions or anomalies that would have precluded normal flight.