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The development of hybrid satellite-terrestrial relay networks (HSTRNs) is one of the driving forces for revolutionizing satellite communications in the modern era. Although there are many unique features of conventional satellite networks, their evolution pace is much slower than the terrestrial wireless networks. As a result, it is becoming more important to use HSTRNs for the seamless integration of terrestrial cellular and satellite communications. With this intent, this paper provides a comprehensive performance evaluation of HSTRNs employing non-orthogonal multiple access technique. The terrestrial relay is considered to be wireless-powered and harvests energy from the radio signal of the satellite. For the sake of comparison, both amplify-and-forward (AF) and decode-and-forward (DF) relaying protocols are considered. Subsequently, the closed-form expressions of outage probabilities and ergodic capacities are derived for each relaying protocol. Extensive simulations are performed to verify the accuracy of the obtained closed-form expressions. The results provided in this work characterize the outage and capacity performance of such a HSTRN.