This paper investigates the secrecy performance of a cognitive relay non-orthogonal multiple access system based on an unmanned aerial vehicle (UAV). In particular, a UAV works as an aerial secondary relay to assist in securing transmissions from a terrestrial secondary source to secondary users in the presence of terrestrial eavesdroppers by sharing the licensed wireless spectrum designated for terrestrial primary systems. The minimum average secrecy rate (MASR) of the secondary users is taken as the main criterion to evaluate the security of the system under the assumption that the location information of the eavesdroppers and successive interference cancellations are imperfect. To maximize MASR, an efficient iterative algorithm is proposed to optimize key system parameters while keeping interference powers within tolerable limits. Since the total MASR problem is non-convex, the successive convex approximation technique is applied to find efficient approximate solutions. Simulation results demonstrate that the proposed algorithm substantially improves the MASR of the system in comparison to benchmark schemes and the orthogonal multiple access system.