We focus on energy-aware path planning of a small unmanned aerial vehicle-based relay which forwards the data received from a stationary remote station to a mobile access point. In order to reduce the communication power required for reliable communication between the unmanned aerial vehicle-based relay and access point, the unmanned aerial vehicle-based relay keeps track of the access point within a maximum allowable horizontal distance at a constant altitude, and thus the communication power of the unmanned aerial vehicle-based relay decreases as the horizontal distance decreases. In order to determine the energy-efficient manoeuvring of the unmanned aerial vehicle-based relay within the horizontal distance, we formulate a new off-line trajectory optimization problem. The solution of the problem gives insight into how to realize the energy-efficient manoeuvring. For online implementation of energy-efficient path planning (manoeuvring control), we propose two control schemes: a model predictive control scheme and a standoff tracking control scheme. These schemes can significantly reduce manoeuvring energy compared with conventional schemes.