We consider a secure communication network, where an unmanned aerial vehicle (UAV) is deployed to transmit short packets of information to multiple ground users (GUs) in the presence of a potential eavesdropper. We provide a fairness-aware design via jointly optimizing the UAV trajectory and user scheduling aiming at maximizing the achievable secrecy rate of the worst-case GU. As the flight time of the UAV is continuous, the non-convex problem involves infinite variables. To deal with this challenge, we introduce the successive-hover-fly (SHF) structure and reconstruct the UAV trajectory with several hovering points, which renders the optimization variable values becoming finite. However, due to the complicated and unanalyzable secrecy rate in the finite blocklength (FBL) regime, the problem remains non-convex and difficult to be efficiently addressed. By constructing a lower-bound concave function at the feasible points in each iteration, we transform the secrecy constraint into a convex one and propose an efficient iterative algorithm. Finally, we present numerical results and validate the effectiveness of our proposed scheme by comparing it with the benchmark.