In this paper, we design the UAV trajectory to minimize the total energy consumption while satisfying the requested timeout (RT) requirement and energy budget, which is accomplished via jointly optimizing the path and UAV's velocities along subsequent hops. The corresponding optimization problem is difficult to solve due to its non-convexity and combinatorial nature. To overcome this difficulty, we solve the original problem via two consecutive steps. Firstly, we propose two algorithms, namely heuristic search, and dynamic programming (DP) to obtain a feasible set of paths without violating the GU's RT requirements based on the traveling salesman problem with time window (TSPTW). Then, they are compared with exhaustive search and traveling salesman problem (TSP) used as reference methods. While the exhaustive algorithm achieves the best performance at a high computation cost, the heuristic algorithm exhibits poorer performance with low complexity. As a result, the DP is proposed as a practical trade-off between the exhaustive and heuristic algorithms. Specifically, the DP algorithm results in near-optimal performance at a much lower complexity. Secondly, for given feasible paths, we propose an energy minimization problem via a joint optimization of the UAV's velocities along subsequent hops. Finally, numerical results are presented to demonstrate the effectiveness of our proposed algorithms. The results show that the DP-based algorithm approaches the exhaustive search's performance with a significantly reduced complexity. It is also shown that the proposed solutions outperform the state-of-the-art benchmarks in terms of both energy consumption and outage performance.