This paper solved a propulsion energy minimization problem subject to a total data-bit constraint of an unmanned aerial vehicle (UAV)-enabled full-duplex mobile relaying system, where a fixed-wing UAV is dispatched as a full-duplex mobile relay to assist data transfer from a source to a destination. Here, three trajectory flying modes are used, namely, the UAV first flies along a circular path above the source, next flies to the destination along a straight line, and finally flies along a circular path above the destination until all the data-bits have been transferred. Since the propulsion energy minimization problem is a nonconvex mixed integer programming problem, its closed-form solution is not tractable, and hence, it is transformed to three subproblems so as to simplify its solution. After solving the three subproblems, an iterative algorithm is proposed to achieve a suboptimal solution to the propulsion energy minimization problem, leading to a new hybrid circular/straight trajectory (HCST) design. Computer simulations are conducted to validate the effectiveness of the proposed HCST design. It is shown that the proposed HCST design performs well in terms of energy saving for the long distance and big data communication cases compared with the straight or circular flight trajectory design. [ABSTRACT FROM AUTHOR]