We propose an adaptive time-delayed photonic reservoir computing (RC) structure by utilizing the Kalman filter (KF) algorithm as training approach. Two benchmark tasks, namely the Santa Fe time-series prediction and the nonlinear channel equalization, are adopted to evaluate the performance of the proposed RC structure. The simulation results indicate that with the contribution of adaptive KF training, the prediction and equalization performance for the benchmark tasks can be significantly enhanced, with respect to the conventional RC using a training approach based on the least-squares (LS). Moreover, by introducing a complex mask derived from a bandwidth and complexity enhanced chaotic signal into the proposed RC, the performance of prediction and equalization can be further improved. In addition, it is demonstrated that the proposed RC system can provide a better equalization performance for the parameter-variant wireless channel equalization task, compared with the conventional RC based on LS training. The work presents a potential way to realize adaptive photonic computing.