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In this paper we propose a novel power adapted network coding (PANC) for a non-orthogonal multiple-access relay channel (MARC), where two sources transmit their information simultaneously to the destination with the help of a relay. Different from the conventional XOR-based network coding (CXNC), the relay in our PANC generates network coded bits by considering the coefficients of the source-to-relay channels, and forwards each bit with a pre-optimized power level. Specifically, by defining a symbol pair as two symbols from the two sources, we first derive the exact symbol pair error rate (SPER) of the system. Noting that the generations of the exact SPER are complicated due to the irregularity of the decision regions caused by random channel coefficients, we propose a coordinate transform (CT) method to simplify the derivations of the SPER. Next, we prove that with a power scaling factor at relay, our PANC scheme can achieve full diversity gain, i.e., two-order diversity gain, of the system, while the CXNC can only achieve one-order diversity gain due to multi-user interference. In addition, we optimize the power levels at the relay to equivalently minimize the SPER at the destination concerning the relationship between SPER and minimum Euclidean distance of the received constellation. Simulation results show that (1) the SPER derived based on our CT method can well approximate the exact SPER with a much lower complexity; (2) the PANC scheme with power level optimizations and power scaling factor design can achieve full diversity, and obtain a much higher coding gain than the PANC scheme with randomly chosen power levels.