Wireless Rechargeable Sensor Networks (WRSNs) with perpetual network lifetime have been used in many Internet of Things (IoT) applications, like oceanic monitoring and precision agriculture. Rechargeable sensors, together with an Unmanned Aerial Vehicle (UAV), are collaboratively employed for fulfilling periodic coverage missions. However, traditional coverage solutions are normally based on static deployment of sensors and not suitable for such novel coverage requirements. In this paper, we propose the concept of Period-Area Coverage (PAC) problem, which requires the data of the overall area must be collected/monitored periodically. To solve the PAC problem, we employ a UAV that simultaneously acts as a mobile charger and sensor. It is responsible for charging nearly exhausted sensors and sensing vacant regions to realize complete event monitoring. To maximize the energy efficiency of the UAV, we propose a heuristic hexagon-based scheduling algorithm (HSA) which can also balance energy consumption. Furthermore, we develop an emergent node charging scheduling method to prevent node exhaustion, and introduce a grid-based boustrophedon scheduling algorithm (GBSA) to reduce the complexity. Finally, we present a charging re-allocation mechanism to further enhance energy efficiency. Extensive simulations demonstrate that the proposed schemes can solve the PAC problem and enhance energy efficiency by at least 18.2% compared to prior arts. Test-bed experiments conducted both in agriculture and oceanic monitoring applications validate the applicability of the proposed scheme in practical scenarios.