This paper investigates a wirelessly-powered mobile edge computing (MEC) framework with the cooperation between an unmanned aerial vehicle (UAV) and a center Cloud. In this system, we minimize the waiting delay of user equipments (UEs) under controllable energy consumption for a UAV serving a large area, where the UAV is equipped with a MEC server and an energy transmitter (ET). Thus, it can provide energy and computing services for UEs and transmit intractable tasks to the Cloud for further execution. To make the UAV serve all UEs, we propose the method by using long and short time slots mixed mechanism, and handle the offloading decision and flying area selection by a mixed-strategy with the Stackelberg game architecture during long time slots, while the resource allocation and trajectory optimization issues are settled within short time slots. Specifically, considering limited battery capacity and severe energy propagation loss of the wireless power transfer (WPT), energy consumption can be controlled through queue optimization in short time slots. We decompose the original nonconvex problem into a series of deterministic optimization sub-problems based on the Lyapunov optimization theory. Furthermore, using Lagrangian duality theory, alternative optimization and successive convex approximation (SCA) technique, we can obtain the closed-form solutions to the original problem. Performance analysis and simulation results demonstrate the convergence and the effectiveness of our proposed algorithm.