Cooperative non-orthogonal multiple access (CNOMA) has been recently adapted with energy harvesting (EH) to increase energy efficiency and extend the lifetime of energy-constrained wireless networks. In EH, two protocols are usually used for CNOMA, namely, power splitting (PS) and time switching (TS). In order to improve the existing EH protocols, this paper investigates a hybrid EH protocol-assisted CNOMA, which is a combination of the two aforementioned main existing EH protocols. The end-to-end bit error rate (BER) and outage probability (OP) expressions of users in our scheme are derived over Nakagami-m fading channels. For the sake of comparison and to highlight the achievable gains, the performance of the hybrid EH (HEH) protocol is compared with the benchmark schemes (i.e., existing EH protocols and no-EH). Based on extensive simulations, it is shown that the analytical results match perfectly with simulations which demonstrates the correctness of the derivations. Numerical results clearly indicate the superiority of the proposed scheme over all the considered benchmarks. In addition, we discuss the optimum value of EH factors to minimize the error and outage in the proposed scheme and show that an optimum value can be obtained according to channel parameters.