In this paper, throughput achieved in cognitive radio channels with finite blocklength codes under buffer limitations is studied. Cognitive users first determine the activity of the primary users' through channel sensing and then initiate data transmission at a power level that depends on the channel sensing decisions. It is assumed that finite blocklength codes are employed in the data transmission phase. Hence, errors can occur in reception and retransmissions can be required. Primary users' activities are modeled as a two-state Markov chain and an eight-state Markov chain is constructed in order to model the cognitive radio channel. Channel state information (CSI) is assumed to be perfectly known by either the secondary receiver only or both the secondary transmitter and receiver. In the absence of CSI at the transmitter, fixed-rate transmission is performed whereas under perfect CSI knowledge, for a given target error probability, the transmitter varies the rate according to the channel conditions. Under these assumptions, throughput in the presence of buffer constraints is determined by characterizing the maximum constant arrival rates that can be supported by the cognitive radio channel while satisfying certain limits on buffer violation probabilities. Tradeoffs between throughput, buffer constraints, coding blocklength, and sensing duration for both fixed-rate and variable-rate transmissions are analyzed numerically. The relations between average error probability, sensing threshold and sensing duration are studied in the case of variable-rate transmissions.