In this letter, we focus on a downlink wireless network aided by an unmanned aerial vehicle (UAV), which has to be recharged periodically at a fixed depot before providing communication service to ground users (GUs), and aim at maximizing the minimum average rate among all GUs over each cycle by jointly optimizing the UAV's time allocation between recharging and service, flight trajectory and transmit power allocation. Such a problem is formulated by considering the UAV's 3-D trajectory design and a practical model for its mechanical energy consumption. Due to the non-convex nature of the original problem, we employ block coordinate decent and successive convex optimization techniques by decomposing it into two sub-problems, which are alternately solved to iteratively improve the minimum average rate. Results show that the proposed solution always outperforms a baseline. Moreover, with cycle length increasing, the performance gap is enlarged and the proposed solution tends to converge to a trajectory, along which the UAV is able to find proper spots at which to hover.