Cooperative non-orthogonal multiple access (CNOMA) has recently been adapted with energy harvesting (EH) to increase energy efficiency and extend the lifetime of energy-constrained wireless networks. This paper proposes a hybrid EH protocol-assisted CNOMA, which is a combination of the two main existing EH protocols (power splitting (PS) and time switching (TS)). The end-to-end bit error rate (BER) expressions of users in the proposed scheme are obtained over Nakagami- fading channels. The proposed hybrid EH (HEH) protocol is compared with the benchmark schemes (i.e., existing EH protocols and no EH). Based on the extensive simulations, we reveal that the analytical results match perfectly with simulations which proves the correctness of the derivations. Numerical results also show that the HEH-CNOMA outperforms the benchmarks significantly. In addition, we discuss the optimum value of EH factors to minimize the error probability in HEH-CNOMA and show that an optimum value can be obtained according to channel parameters.