This paper analyze a new cooperative hybrid underlay–overlay cognitive radio non-orthogonal multiple access (CHCR-NOMA) system, where a secondary source and a primary source share a secondary decode-and-forward relay by using NOMA principles. Exact closed-form expressions for the outage probability of both primary and secondary destinations are derived assuming imperfect channel state information (CSI) and imperfect successive interference cancellation (SIC). Moreover, the asymptotic expressions of the outage probability at high signal-to-noise ratio (SNR) are also derived. The optimal power allocation coefficients for various transmit powers and different distances of the primary destination from the relay are determined to ensure priority to the primary user or to satisfy the fairness requirements of both the destinations. The paper also investigates the impact of various parameters such as channel estimation error (CEE), detection threshold, and power allocation coefficients on the outage performance of the proposed system. The accuracy of derived analytical results is verified through the Monte Carlo simulations.