This paper introduces hybrid automatic repeat request with incremental redundancy (HARQ-IR) to boost the reliability of short packet communications. The finite blocklength information theory and correlated decoding events tremendously preclude the analysis of average block error rate (BLER). Fortunately, the recursive form of average BLER motivates us to calculate its value through the trapezoidal approximation and Gauss-Laguerre quadrature. Moreover, the asymptotic analysis is performed to derive a simple expression for the average BLER at high signal-to-noise ratio (SNR). Then, we study the maximization of long term average throughput (LTAT) via power allocation meanwhile ensuring the power and the BLER constraints. For tractability, the asymptotic BLER is employed to solve the problem through geometric programming (GP). However, the GP-based solution underestimates the LTAT at low SNR due to a large approximation error in this case. Alternatively, we also develop a deep reinforcement learning (DRL)-based framework to learn power allocation policy. In particular, the optimization problem is transformed into a constrained Markov decision process, which is solved by integrating deep deterministic policy gradient (DDPG) with subgradient method. The numerical results finally demonstrate that the DRL-based method outperforms the GP-based one at low SNR, albeit at the cost of increasing computational burden.