This paper addresses the issue of network disconnection in UAV networks caused by the departure of multiple drones due to energy depletion or damage during flight. We propose an energy-constrained topology recovery algorithm based on Graph Convolutional Networks (GCN). A network topology connectivity model is developed, incorporating an energy constraint module, with the optimization objectives of rapid self-healing and minimized energy consumption. Introducing a novel weighted loss function to balance the trade-off between minimizing energy consumption and movement distance. In the event of a network disconnection, the remaining energy, and link connectivity information of the operational UAVs are encoded into a Laplacian matrix, which is then fed into the GCN model as the input. The model is trained using gradient descent, resulting in a UAV flight strategy for topology recovery. Simulation results demonstrate that, compared to the currently known methods, the proposed approach reduces self-healing time by 41.2%, overall network communication time by 13.3%, and improves the self-healing rate by 27%. Furthermore, the proposed method achieves effective energy control while maintaining high network stability and persistence.