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We consider a fully-loaded ground wireless network supporting unmanned aerial vehicle (UAV) transmission services. To enable the overload transmissions to a ground user (GU) and a UAV, two transmission schemes are employed, namely non-orthogonal multiple access (NOMA) and relaying, depending on whether or not the GU and UAV are served simultaneously. Under the assumption of the system operating with infinite blocklength (IBL) codes, the IBL throughputs of both the GU and the UAV are derived under the two schemes. More importantly, we also consider the scenario in which data packets are transmitted via finite blocklength (FBL) codes, i.e., data transmission to both the UAV and the GU is performed under low-latency and high reliability constraints. In this setting, the FBL throughputs are characterized again considering the two schemes of NOMA and relaying. Following the IBL and FBL throughput characterizations, optimal resource allocation designs are subsequently proposed to maximize the UAV throughput while guaranteeing the throughput of the cellular user.Moreover, we prove that the relaying scheme is able to provide transmission service to the UAV while improving the GU's performance, and that the relaying scheme potentially offers a higher throughput to the UAV in the FBL regime than in the IBL regime. On the other hand, the NOMA scheme provides a higher UAV throughput (than relaying) by slightly sacrificing the GU's performance.