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A design of a new powerful high-frequency short-pulse coaxial-cavity harmonic gyrotron operating at first, second, and third harmonic is presented. The design includes the analysis of mode competition between the fundamental and higher harmonic modes, as well as the investigation of the possibility of the corresponding fast frequency tuning of the gyrotron by changing the accelerating voltage. The possibilities of modification of the existing coaxial cavity gyrotron operating at present at the Forschungszentrum Karlsruhe (FZK) for operation at the second harmonic are discussed. Potentialities of such gyrotrons for plasma physics applications are noted.