On May 29, 2020, at 1715 central daylight time, a Magni M-16 gyroplane, N316MG, was substantially damaged when it was involved in an accident near Cape Girardeau, Missouri. The student pilot received minor injuries. The aircraft was operated as a Title 14 Code of Federal Regulations (CFR) Part 91 instructional flight. The student pilot was conducting a supervised solo flight to practice touch-and-go landings and stated that, while on approach for the fourth or fifth landing, the engine warning light illuminated while on the downwind leg. He dismissed the engine light when it extinguished while rolling down the runway and elected to take off again. He then noticed a “weird smell” after he took off and the engine did not seem to be producing power, so he landed again; however, because the aircraft was nearing the end of the runway, he chose to take off again so that the gyroplane did not continue off the end of the runway into the muddy grass. The pilot reported that, after takeoff, he “had no speed” and “fell behind the power curve,” and the gyroplane descended into the grass past the end of the runway. The wheels dug into the soft ground and the aircraft rolled onto its side, substantially damaging the rotor system, the airframe, and the tail assembly. The flight instructor and owner of the gyroplane watched the pilot land and stated that the pilot remained on the runway too long before taking off again. He stated that he was expecting the pilot to exit the runway and taxi back when he took off again with about 100 to 200 ft of runway remaining. The nose of the gyroplane was “too high,” and the gyroplane settled back to the ground past the departure end of the runway. He explained that the accident gyroplane was equipped with an engine alarm light that would illuminate for high exhaust gas temperatures (EGT), low oil pressure, low fuel pressure, and low voltage. He believed that the light illuminated on downwind for high EGT due to having the throttle at full power for too long, and that the light went out when the throttle was reduced for landing. He further explained that the light likely illuminated on landing due to low oil pressure when the throttle was reduced to idle. The engine was examined by a local mechanic and no anomalies were noted that would have precluded normal operation.