In this study, the authors examine the performance of an uplink non-orthogonal multiple access (NOMA) system with cooperative full-duplex relaying (CFR-NOMA), where the user closer to the base station (BS) is considered as a full-duplex relay to aid the transmission from the other user to the BS. The closer user first decodes the signals transmitted from the far user and then forwards them using superposition coding to the BS on top of transmitting its own information signals to the BS. First, they analyse the performance of the CFR-NOMA system in terms of the outage probability and average sum rate. Then, they analytically obtain the optimal power allocation coefficients that minimise the outage probability. They also study the optimal power allocation to maximise the average sum rate under specified constraints. Their examination demonstrates that the CFR-NOMA system can significantly outperform the uplink conventional orthogonal multiple access and uplink NOMA with half-duplex relaying system in terms of achieving a lower outage probability or a higher average sum rate.