In this paper, we investigate an unmanned aerial vehicle (UAV)-enabled covert communication network with multiple ground users (GUs), a warden, and a UAV. The UAV carries directional antenna to communicate with GUs covertly, i.e., without being exposed to the warden. Focusing on task to improve the throughput of all GUs, we provide a joint user assignment and trajectory design aiming to maximize the throughput of GUs with worst condition. In particular, we consider the impact of directional antenna pattern on covertness performance, and derive the expression for maximum allowed transmit power satisfying the covertness and maximum transmit power requirement based on the modified antenna pattern. Following the characterization, the non-convex joint design problem is formulated. For making the highly non-convex problem analysable, we adopt two lemmas to construct the concave approximation function for the throughput which enables us to build up a convex problem. Accordingly, an effectively iterative algorithm is putting forward to settle the problem. Finally, numerical results confirm that our scheme with directional antenna outperforms the benchmarks.