Unmanned aerial vehicles (UAVs) and Terahertz (THz) technology are envisioned to play paramount roles in next-generation wireless communications. In this paper, we present a novel secure UAV-assisted mobile relaying system operating at THz bands for data acquisition from multiple ground user equipments (UEs) towards a destination. We assume that the UAV-mounted relay may act, besides providing relaying services, as a potential eavesdropper called the untrusted UAV-relay (UUR). To safeguard end-to-end communications, we present a secure two-phase transmission strategy with cooperative jamming. Then, we devise an optimization framework in terms of a new measure $-$ secrecy energy efficiency (SEE), defined as the ratio of achievable average secrecy rate to average system power consumption, which enables us to obtain the best possible security level while taking UUR's inherent flight power limitation into account. For the sake of quality of service fairness amongst all the UEs, we aim to maximize the minimum SEE (MSEE) performance via the joint design of key system parameters, including UUR's trajectory and velocity, communication scheduling, and network power allocation. Since the formulated problem is a mixed-integer nonconvex optimization and computationally intractable, we decouple it into four subproblems and propose alternative algorithms to solve it efficiently via greedy/sequential block successive convex approximation and non-linear fractional programming techniques. Numerical results demonstrate significant MSEE performance improvement of our designs compared to other known benchmarks.