In this paper, we consider an energy-constrained unmanned aerial vehicle (UAV)-enabled mobile relay assisted secure communication system in the presence of a legitimate source-destination pair and multiple eavesdroppers with imperfect locations. The energy-constrained UAV employs the power splitting (PS) scheme to simultaneously receive information and harvest energy from the source, and then exploits the time switching (TS) protocol to perform information relaying. Furthermore, we consider a full-duplex destination node which can simultaneously receive confidential signals from the UAV and cooperatively transmit artificial noise (AN) signals to confuse malicious eavesdroppers. To further enhance the reliability and security of this system, we formulate a worst case secrecy rate maximization problem, which jointly optimizes the position of the UAV, the AN transmit power, as well as the PS and TS ratios. The formulated problem is non-convex and generally intractable. In order to circumvent the non-convexity, we decouple the original optimization problem into three subproblems; this facilitates the design of a suboptimal iterative algorithm. In each iteration, we propose a multi-dimensional search and numerical method to handle the subproblem. Numerical simulation results are provided to demonstrate the effectiveness and superior performance of the proposed joint design versus the conventional schemes in the literature.