We have measured the flexophotovoltaic effect of single crystals of halide perovskites MAPbBr_{3} and MAPbI_{3}, as well as the benchmark oxide perovskite SrTiO_{3}. For halide perovskites, the flexophotovoltaic effect is found to be orders of magnitude larger than for SrTiO_{3}, and indeed large enough to induce photovoltages bigger than the band gap. Moreover, we find that in MAPbI_{3} the flexophotovoltaic effect is additional to a native bulk photovoltaic response that is switchable and ferroelectric-like. The results suggest that strain gradient engineering can be a powerful tool to modify the photovoltaic output even in already well-established photovoltaic materials.