An unmanned aerial vehicle (UAV)-assisted mobile edge computing (MEC) networks fulfil the on-demand computation services for the mobile terminals (MTs) with its high mobility and easy deployment. Thus, reduction in the latency is observed but the energy efficiency is still a major issue in this network, as both the UAV and MTs have limited energy storage batteries. Also, to support the massive connectivity, and to handle the data traffic generated by the MTs in an uplink scenario, using nonorthogonal multiple access (NOMA) is accomodated. The NOMA-based MEC provides flexible computing services to MTs. In this article, our goal is to minimize the total energy consumption of the NOMA-based MEC networks underlaying UAV with time, computation capacity, and UAV trajectory. The formulated model is a nonlinear programming problem; for the solution, it is divided into two subproblems: joint time allocation and task computation capacity, and UAV trajectory optimization. To solve the joint time allocation and computation task capacity, we proposed an iterative algorithm with low complexity, where closed-form solutions are obtained in each step. Next, to optimize the trajectory of the UAV, we used the successive convex approximation technique. Numerical results demonstrated that the proposed scheme achieves better results as compared to the orthogonal multiple access and equal resource allocation schemes.