This paper investigates system performance in the Internet of Things (IoT) with an energy harvesting (EH) unmanned aerial vehicle (UAV)-enabled relay under Nakagami- m fading, where the time switching (TS) and adaptive power splitting (APS) protocols are applied for the UAV. Our proposed system model consists of a base station (BS), two IoT device (ID) clusters (i.e., a far cluster and a near cluster), and a multiantenna UAV-enabled relay (UR). We adopt a UR-aided TS and APS (U-TSAPS) protocol, in which the UR can dynamically optimize the respective power splitting ratio (PSR) according to the channel conditions. To improve the throughput, the nonorthogonal multiple access (NOMA) technique is applied in the transmission of both hops (i.e., from the BS to the UR and from the UR to the ID clusters). The U-TSAPS protocol is divided into two phases. In the first phase, the BS transmits a signal to the UR. The UR then splits the received signal into two streams for information processing and EH using the APS scheme. In the second phase, the selected antenna of the UR forwards the received signal to the best far ID (BFID) in the far cluster and the best near ID (BNID) in the near cluster using the decode-and-forward (DF) or amplify-and-forward (AF) NOMA scheme. We derive closed-form expressions for the outage probabilities (OPs) at the BFID and BNID with the APS ratio under imperfect channel state information (ICSI) to evaluate the system performance. Based on these derivations, the throughputs of the considered system are also evaluated. Moreover, we propose an algorithm for determining the nearly optimal EH time for the system to minimize the OP. In addition, Monte Carlo simulation results are presented to confirm the accuracy of our analysis based on simulations of the system performance under various system parameters, such as the EH time, the height and position of the UR, the number of UR antennas, and the number of IDs in each cluster.