Microgrids are a crucial component of modern energy internet, offering reliable distributed generation solutions. However, communication failures in data acquisition terminals, caused by long-distance transmission and electromagnetic interference, can lead to data loss and negatively impact economic dispatch accuracy. This study introduces a novel deep reinforcement learning algorithm, the Relative Entropy Regularized Asynchronous Update Soft Actor-Critic algorithm, to address these challenges. First, the algorithm compensates for missing data during communication interruptions using predictive modeling techniques, thereby ensuring continuous and accurate system operation. Then, to handle the challenges posed by prediction errors, a relative entropy regularization strategy is applied during training to maintain policy stability. Finally, upon communication restoration, the algorithm asynchronously updates the reward function using actual historical data to correct cumulative errors introduced by the predictive model. Results demonstrate that this method significantly enhances economic efficiency under photovoltaic and load communication failures.