In this paper, we propose an energy-efficient covert communication framework for unmanned aerial vehicle (UAV)-assisted backscatter systems, where one UAV is deployed to collect data from one ground backscatter device (BD), in the presence of a warden who is trying to detect whether BD is transmitting or not. We first analyze the false alarm probability (FAP) and miss detection probability (MDP) for the covert communication and investigate the detection error probability by considering the location and channel uncertainty of warden. Then, aimed at maximizing the energy efficiency (EE) of our proposed system subject to reliability and covertness constraints, we optimize the UAV transmit power and hovering point jointly and design a low-complexity solution based on alternating optimization (AO), Dinkelbach method and successive convex approximation (SCA) solutions. Finally, simulation results are provided to verify the effectiveness of our proposed algorithm and highlight the fundamental trade-off between the EE and covertness constraint.