We study the throughput-vs-delay trade-off in an overlay multi-channel single-secondary-user cognitive radio system. Due to the limited sensing capabilities of the cognitive radio user, channels are sensed sequentially. Maximizing the throughput in such a problem is well-studied in the literature. Yet, in real-time applications, hard delay constraints need to be considered besides throughput. In this paper, optimal stopping rule and optimal power allocation are discussed to maximize the secondary user's throughput, subject to an average delay constraint. We provide a low complexity approach to the optimal solution of this problem. Simulation results show that this solution allows the secondary user to meet the delay constraint without sacrificing throughput significantly. It also shows the benefits of the optimal power allocation strategy over the constant power allocation strategy.