We report on the observation of cyclotron-resonance-induced photocurrents, excited by continuous wave terahertz radiation, in a three-dimensional topological insulator (TI) based on an 80-nm strained HgTe film. The analysis of the photocurrent formation is supported by complementary measurements of magnetotransport and radiation transmission. We demonstrate that the photocurrent is generated in the topologically protected surface states. Studying the resonance response in a gated sample, we examined the behavior of the photocurrent, which enables us to extract the mobility and the cyclotron mass as a function of the Fermi energy. For high gate voltages, we also detected cyclotron resonance (CR) of bulk carriers, with a mass about two times larger than that obtained for the surface states. The origin of the CR-assisted photocurrent is discussed in terms of asymmetric scattering of TI surface carriers in the momentum space. Furthermore, we show that studying the photocurrent in gated samples provides a sensitive method to probe the cyclotron masses and the mobility of two-dimensional Dirac surface states, when the Fermi level lies in the bulk energy gap or even in the conduction band.