The student sport pilot was flying a passenger on a local area flight when he experienced what he described as a "mushy" spot in the engine at approximately 4,500 revolutions per minute. The student pilot landed in the desert to check the engine's spark plugs. After he verified that the spark plugs were fine and performed another run up with no anomalies noted, the student pilot attempted to takeoff. As the aircraft was gaining ground speed, the student pilot saw trees and a gully quickly approaching. He decided that the aircraft would not be able to clear the trees so he reduced power and applied the brakes. The aircraft impacted the gully, the nose gear broke, and the aircraft pitched down and settled to one side. During the accident sequence, the compression strut was damaged and the mast support was bent. The student pilot indicated that he misjudged the hardness of the soil and the terrain was softer than he initially thought. According to 14 Code of Federal Regulations Part 61.89, a student pilot may not act as pilot-in-command of an aircraft that is carrying a passenger.