Rate Splitting Multiple Access (RSMA) has recently emerged as a promising technique to enhance the transmission rate for multiple access networks. Unlike conventional multiple access schemes, RSMA requires splitting and transmitting messages at different rates. The joint optimization of the power allocation and rate control at the transmitter is challenging given the uncertainty and dynamics of the environment. Furthermore, securing transmissions in RSMA networks is a crucial problem because the messages transmitted can be easily exposed to adversaries. This work first proposes a stochastic optimization framework that allows the transmitter to adaptively adjust its power and transmission rates allocated to users, and thereby maximizing the sum-rate and fairness of the system under the presence of an adversary. We then develop a highly effective learning algorithm that can help the transmitter to find the optimal policy without requiring complete information about the environment in advance. Extensive simulations show that our proposed scheme can achieve positive covert transmission rates in the finite blocklength regime and non-saturating rates at high SNR values. More significantly, our achievable covert rate can be increased at high SNR values (i.e., 20 dB to 40 dB), compared with saturating rates of a conventional multiple access scheme.