In this paper, we study the outage and ergodic rate performance of a rate-splitting multiple access (RSMA)-based cognitive radio (CR) system, where the secondary transmitter aims to communicate with two RSMA users. Imperfect successive interference cancellation (ipSIC) and channel estimation errors (CEEs) are considered in the proposed analysis. Analytical expressions for the outage probability (OP) and the ergodic rate (ER) are calculated. For a deeper understanding, the asymptotic behavior of OP and ER at high signal-to-noise ratio (SNR) regimes are carried out. Illustrative simulation results are presented and reveal that: i) The OP decreases and the ER gradually increases with the increase of SNR, and eventually approaching a constant; ii) ipSIC and CEEs have a negative impact on the considered system; iii) The outage and ergodic rate performance of RSMA-based CR network outperforms CRNOMA network due to flexible power allocation between common and private messages; iv) The OP first decreases and then increases with the power allocated to the common message, and there exists an optimal power allocation factor to ensure the reliable performance.