Unmanned aerial vehicle (UAV) assisted wireless communication is a key component of the next-generation mobile networks. In coping with the increased dynamics in UAV networks, the transmission of control information is indispensable, requiring not only ultra reliability and low latency, but also high security. In this paper, we investigate the secrecy performance of the control information in a NOMA ground-air short-packet wireless network with an untrusted internal UAV or an external flying eavesdropper, respectively. Both the large-scale path loss and the Nakagami-m small-scale fading are considered. First, the closed-form expressions of the average secure block error rate (BLER) and the average achievable secrecy throughput in each scenario are derived. Then, the asymptotic performance in the high signal-to-noise ratio (SNR) regime is analyzed to get more insights from both scenarios. Specifically, analytical results show that error floors occur with the increase of SNR. Moreover, a one-dimensional search is applied to maximize the average achievable secrecy throughput by optimizing the blocklength. Simulation results are provided to verify the accuracy of analysis and the effectiveness of optimization.