Data integrity of power system states is critical to modern power grid operation and control. Due to communication latency, state measurements are not immediately available at the control center, rendering slow responses of time-sensitive applications. In this paper, a new graph-based deep learning approach is proposed to recover and predict the states ahead of time utilizing the power network topology and existing measurements. A graph-convolutional recurrent adversarial network is devised to process available information and extract graphical and temporal data correlations. This approach overcomes drawbacks of the existing synchrophasor recovery and prediction implementation to improve the overall system performance. Additionally, the approach offers an adaptive data processing method to handle power grids of various sizes. Case studies demonstrate the outstanding recovery and prediction accuracy of the proposed approach, and investigations are conducted to illustrate its robustness against bad communication conditions, measurement noise, and system topology changes.