In this paper, we propose a secure short-packet communication (SPC) system involving an unmanned aerial vehicle (UAV)-aided relay in the presence of a terrestrial passive eavesdropper. The considered system, which is applicable to various next-generation Internet-of-Things (IoT) networks, exploits a UAV as a mobile relay, facilitating the reliable and secure exchange of intermittent short packets between a pair of remote IoT devices with strict latency. Our objective is to improve the overall secrecy throughput performance of the system by carefully designing key parameters such as the coding blocklengths and the UAV trajectory. However, this inherently poses a challenging optimization problem that is difficult to solve optimally. To address the issue, we propose a low-complexity algorithm inspired by the block successive convex approximation approach, where we divide the original problem into two subproblems and solve them alternately until convergence. Numerical results demonstrate that the proposed design achieves significant performance improvements relative to other benchmarks, and offer valuable insights into determining appropriate coding blocklengths and UAV trajectory.