We propose and experimentally demonstrate a photonic time-delay reservoir computing (TDRC) system with random distributed optical feedback under optical injection. To evaluate the performance, we calculate the memory ability and perform two benchmark tasks, i.e., chaotic time series prediction and nonlinear channel equalization task. Our numerical results show that the proposed TDRC has a superior performance compared with the case with conventional single optical feedback. This is attributed to the fact that the random distributed optical feedback offers multiple external cavity modes, which enhance the nonlinearity of the reservoir laser. Additionally, the experimental result also shows that our proposed TDRC scheme outperforms the computer with single optical feedback in the chaotic time series prediction task. To the best of our knowledge, our work offers a novel path to improve the performance of TDRC by introducing random distributed optical feedback.