The access of massive Internet of Things (IoT) users poses several challenges for Unmanned Aerial Vehicle (UAV)-aided communications, particularly in terms of security and reliability. This paper investigates a secure and robust power allocation scheme for UAV-aided IoT Non-Orthogonal Multiple Access (NOMA) downlink networks with a potential eavesdropper, considering imperfect Channel State Information (CSI). Given the noise uncertainty caused by the UAV’s mobility and the statistical channel estimation error, we formulate a robust optimization problem to maximize the total covert rate of all NOMA users, subject to covertness and rate-based reliability constraints. To solve this optimization problem, we first derive the minimum detection error rate and utilize the statistical characteristics (i.e., the mean and variance of channel gain errors) to obtain the deterministic covertness and reliability constraints, respectively. We then prove that the problem is concave and determine the optimal power allocation algorithm using the Karush–Kuhn–Tucker conditions. Extensive numerical simulations validate the effectiveness of the proposed algorithm and demonstrate its ability to realize more secure and robust UAV-aided IoT systems.