The problem of multi-robot coverage control becomes significantly challenging when multiple robots leave the mission space simultaneously to charge their batteries, disrupting the underlying network topology for communication and sensing. To address this, we propose a resilient network design and control approach that allows robots to achieve the desired coverage performance while satisfying energy constraints and maintaining network connectivity throughout the mission. We model the combined motion, energy, and network dynamics of the multirobot systems (MRS) as a hybrid system with three modes, i.e., coverage, return-to-base, and recharge, respectively. We show that ensuring the energy constraints can be transformed into designing appropriate guard conditions for mode transition between each of the three modes. Additionally, we present a systematic procedure to design, maintain, and reconfigure the underlying network topology using an energy-aware bearing rigid network design, enhancing the structural resilience of the MRS even when a subset of robots departs to charge their batteries. Finally, we validate our proposed method using numerical simulations.