An energy-efficient two-way (TW) full-duplex (FD) network with the assistance of an unmanned aerial vehicle (UAV) is proposed, where the UAV acts as a mobile relay to assist the information exchange between two terrestrial transceivers. In particular, the self-interference (SI) channel gains follow complex Gaussian distribution and the perfect channel state information (CSI) of SI channels is unavailable at the receiver. To maximize the energy efficiency (EE), UAV flight speeds are controlled and power adaptation at the UAV relay is performed. The genetic algorithm (GA) is applied to efficiently obtain the optimal solution. Numerical results show that our scheme performs better than the one-way (OW) FDR scheme, fixed power (FP) and fixed flight speed (FS) policy. In addition, the SI cancellation factor on the EE is also demonstrated.