Mobile edge computing (MEC) has been considered as a promising technique to address the explosively growing computation-intensive applications. Thanks to the flexibility of the unmanned aerial vehicles (UAVs), the UAV-assisted MEC can serve mobile terminals (MTs) effectively, i.e., the computing server installed on the UAV can flexibly change its location to serve MTs. Moreover, since non-orthogonal multiple access (NOMA) is able to accommodate massive connectivity, the NOMA-based and UAV-assisted MEC can provide flexible computing services for MTs in large-scale access networks (e.g., sensor networks and Internet of Things). However, due to the diversity of the UAV's trajectory and the interference among MTs introduced by NOMA, the performance (e.g., energy consumption and delay) of the NOMA-based and UAV-assisted MEC system is adversely affected. Therefore, in this paper, we formulate an optimization problem to minimize the largest energy consumption among MTs by jointly optimizing the trajectory, task data and computing resource allocations, and then propose an iterative algorithm to solve the optimization problem. Furthermore, to minimize the largest energy consumption among MTs with lower complexity, we propose a fixed point service scheme and optimize the location of the fixed point. The simulation results show that the proposed optimization algorithms can effectively reduce the largest energy consumption among MTs and ensure the fairness among MTs.