We investigate the high-energy (<1 GeV) emission from the pulsar PSR B1509-58, and its relation to the radio emission in the 1.4-GHz frequency band. We investigate the role that quasi-linear diffusion plays in the production of pulsed high-energy radiation. We show that the relatively low-frequency waves are excited by cyclotron instability and, because of the diffusion process, this influences the particle distribution function and switches on the synchrotron emission mechanism. We argue that the coincidence of the high-energy main peak and the radio pulse is a direct consequence of the fact that both high- and low-frequency radiation are produced simultaneously in the local area of the pulsar magnetosphere. We also consider and explain the absence of the radio counter pulse.