This paper investigates the resource management problem in multi-carrier rate-splitting multiple access (MC-RSMA) systems with imperfect channel state information (CSI) and successive interference cancellation (SIC) for ultra-reliable and low-latency communications (URLLC) applications. To explore the trade-off between the decoding error probability and achievable rate, effective throughput (ET) is adopted as the utility function in this study. Then, a mixed-integer non-convex problem is formulated, where power allocation, rate control, and user grouping are jointly taken into consideration. To tackle this problem, we approximate the achievable ET using a lower bound and then develop a decomposition method to decouple optimization variables. Specifically, for a given user grouping scheme, an iteration-based concave-convex programming (CCCP) method and an iteration-free lower-bound approximation (LBA) method are proposed for power allocation and rate control. Next, a greedy search-based scheme and a heuristic grouping scheme are developed for the user-grouping problem. The simulation results verify the effectiveness of the CCCP and LBA methods in power allocation and rate control and the greedy search-based and heuristic grouping methods in user grouping. Besides, the superiority of RSMA for URLLC services is demonstrated when compared to spatial division multiple access.