We present a calculation of inelastic $J/\ensuremath{\psi}$ production in polarized photon-hadron collisions. We consider the photoproduction of $J/\ensuremath{\psi}$ in the energy range relevant to the DESY HERA collider. The Weizs\"acker-Williams approximation is adopted in the evaluation of the cross sections for $\mathrm{ep}$ collisions. We find that this process can give another independent test for the color-octet mechanism, and the different features for the two color-octet processes may provide further information on the mechanism for inelastic $J/\ensuremath{\psi}$ photoproduction. The discrepancy in the production asymmetry $A$ between various sets of polarized gluon distribution functions is also found to be distinctive.