In this paper, throughput achieved in cognitive radio channels with finite blocklength codes in the presence of buffer constraints is studied. It is assumed that cognitive secondary users initially perform channel sensing in order to determine the activity of the primary users. Following channel sensing, secondary transmitter sends the data at fixed power and rate whose values depend on the channel sensing decisions. Data transmissions are assumed to be performed with finite blocklength codes. Hence, errors can occur in reception and retransmissions can be required. Under these assumptions and a Markov model for primary user activity, a state-transition model for the cognitive radio channel is constructed. For this model, throughput under 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 are analyzed numerically.