With the advantages of maneuverability and low cost, unmanned aerial vehicles (UAVs) are widely deployed in mobile-edge computing (MEC) as micro servers to provide computing service. However, tasks usually require a large amount of energy and have strict time constraints, while the battery energy and endurance of UAVs are limited. Therefore, energy consumption and delay have become key issues in such architectures. To address this issue, an entropy normalized soft actor-critic (ENSAC) computation offloading algorithm is proposed in this article, aiming to minimize the weighted sum of task offloading delay and energy consumption. In ENSAC, we formulate the task offloading problem as a Markov decision process (MDP). Considering the nonconvexity, high-dimensional state space, and continuous action space of this problem, the ENSAC algorithm fully combines deviation strategy and maximum entropy reinforcement learning and designs a system utility function under entropy normalization as a reward function, thus ensuring fairness in weighted energy consumption and delay. What is more, the ENSAC algorithm also considers UAV trajectory planning, task offloading ratio, and power allocation in the UAV-assisted MEC system. Therefore, compared with previous methods, the ENSAC algorithm has stronger stability, better exploration performance, and can handle more complex environments and larger action space. Finally, extensive experiments demonstrate that, in both energy-saving and delay-sensitive scenarios, the ENSAC algorithm can quickly converge to the optimal solution while maintaining stability. Compared with four benchmark algorithms, it reduces the total system cost by 52.73%.