To alleviate the strict synchronization requirements and avoid the high peak-to-average power issue, asynchronous cyclic prefixed single carrier (CP-SC) transmission is applied for an uplink NOMA communication system. In this paper, we investigate rate splitting (RS) to improve the outage performance of the considered system. Two RS schemes, namely, the fixed RS and cognitive RS schemes, are proposed under back-off power control. With the aid of frequency-domain equalization, successive interference cancellation (SIC) is applied at the base station for recovering the desired signal. Under frequency-selective fading channels, closed-form expressions for the outage probabilities are derived for both near and far users achieved by asynchronous CP-SC transmission with and without RS, respectively. Motivated by the cognitive radio concept, the cognitive RS scheme implements the power allocation to sequentially guarantee the outage performance of the virtual primary, secondary, and tertiary users according to the SIC processing order. The enhanced outage performance achieved by the fixed RS and cognitive RS schemes are corroborated by Monte Carlo simulation results.