Ferroelectric materials are interesting candidates for future photovoltaic applications due to their potential to overcome the fundamental limits of conventional single bandgap semiconductor-based solar cells. Although a more efficient charge separation and above bandgap photovoltages are advantageous in these materials, tailoring their photovoltaic response using ferroelectric functionalities remains puzzling. Here we address this issue by reporting a clear hysteretic character of the photovoltaic effect as a function of electric field and its dependence on the poling history. Furthermore, we obtain insight into light induced nonequilibrium charge carrier dynamics in Bi2FeCrO6 films involving not only charge generation, but also recombination processes. At the ferroelectric remanence, light is able to electrically depolarize the films with remanent and transient effects as evidenced by electrical and piezoresponse force microscopy (PFM) measurements. The hysteretic nature of the photovoltaic response and its nonlinear character at larger light intensities can be used to optimize the photovoltaic performance of future ferroelectric-based solar cells.