Thanks to the benefits of non-orthogonal multiple access (NOMA) in wireless communications, we evaluate a wireless sensor network deploying NOMA (WSN-NOMA), where the destination can receive two data symbols in a whole transmission process with two time slots. In this work, two relaying protocols, so-called time-switching-based relaying WSN-NOMA (TSR WSN-NOMA) and power-splitting-based relaying WSN-NOMA (PSR WSN-NOMA) are deployed to study energy-harvesting (EH). Regarding the system performance analysis, we obtain the closed-form expressions for the exact and approximate outage probability (OP) in both protocols, and the delay-limited throughput is also evaluated. We then compare the two protocols theoretically, and two optimization problems are formulated to reduce the impact of OP and optimize the data rate. Our numerical and simulation results are provided to prove the theoretical and analytical analysis. Thanks to these results, a great performance gain can be achieved for both TSR WSN-NOMA and PSR WSN-NOMA if optimal values of TS and PS ratios are found. In addition, the optimized TSR WSN-NOMA outperforms that of PSR WSN-NOMA in terms of OP.