We propose a hole-induced mechanism of spin-polarized current generation by circularly polarized synchrotron radiation and corresponding induced magnetization in magnetically-doped topological insulators Bi$_{1.37}$V$_{0.03}$Sb$_{0.6}$Te$_2$Se. Considered spin-polarized current is generated due to the spin-dependent depopulation of the Dirac cone topological surface states at the Fermi level and subsequent compensation of the generated holes. We have found experimentally and theoretically a relation between the generated spin-polarized current and the shift of the electrochemical potential. The out-of-plane magnetization induced by circularly polarized synchrotron radiation and its inversion with switching the direction of circular polarization were experimentally shown and theoretically confirmed.