Damping of small-scale fast magnetosonic waves (FMSWs) in large tokamak plasmas due to cyclotron absorption by a small group of resonance alpha particles (products of the thermonuclear fusion) is determined on the basis of a nonlocal approach. Such a nonlocal approach for the separate modes of a FMSW gives the same expression for the damping coefficient (except for special cases) as obtained in the local approach, although the distribution of high-frequency fields in the cyclotron resonance range may differ greatly. The cyclotron absorption of these waves by alpha particles may be essential, and it competes with electron Cherenkov absorption and cyclotron absorption by ions of other species during plasma heating and maintenance of the current in reactor tokamaks by means of FMSWs.