In research endeavors focused on the advancements of sixth-generation wireless communications, there is an urgent need to explore more efficient and flexible multiple-access schemes. Additionally, there is a growing demand for methods that can effectively conceal communication behavior, a necessity that is particularly crucial for security-critical applications. This study investigates the covertness and effective covert throughput (ECT) of uplink rate-splitting multiple access (RSMA) systems with overt and covert users in the finite block length regime. Through the derivation of the detection error probability for minimum error probability detection and the missed detection probability for Neyman-Pearson detection, it is demonstrated that the considered uplink RSMA scheme surpasses time division multiple access and frequency division multiple access counterparts with regard to covertness. Subsequently, an iterative optimization framework based on successive convex approximation is proposed to maximize the average amount of information reliably and covertly transmitted within the block length, namely, the ECT. Furthermore, the scenario where the covert user sends both covert and overt messages using RSMA is examined, and it is shown that the covert user can achieve higher effective throughput than the other schemes.