In this study, we design and analyze a reliability-oriented downlink wireless network assisted by unmanned aerial vehicles (UAVs). This network employs non-orthogonal multiple access (NOMA) transmission and finite blocklength (FBL) codes. In the network, ground user equipments (GUEs) request content from a remote base station (BS), and there are no direct connections between the BS and the GUEs. To address this, we employ a UAV with a limited caching capacity to assist the BS in completing the communication. The UAV can either request uncached content from the BS and then serve the GUEs or directly transmit cached content to the GUEs. In this paper, we first introduce the decoding error rate within the FBL regime and explore caching policies for the UAV. Subsequently, we formulate an optimization problem aimed at minimizing the average maximum end-to-end decoding error rate across all GUEs while considering the coding length and maximum UAV transmission power constraints. We propose a two-step alternating optimization scheme embedded within a deep deterministic policy gradient (DDPG) algorithm to jointly determine the UAV trajectory and transmission power allocations, as well as blocklength of downloading phase, and our numerical results show that the combined learning-optimization algorithm efficiently addresses the considered problem. In particular, it is shown that a well-designed UAV trajectory, relaxing the FBL constraint, increasing the cache size, and providing a higher UAV transmission power budget all lead to improved performance.