In this paper, we propose a supervised single-channel speech enhancement method that combines Kullback-Leibler (KL) divergence-based non-negative matrix factorization (NMF) and a hidden Markov model (NMF-HMM). With the integration of the HMM, the temporal dynamics information of speech signals can be taken into account. This method includes a training stage and an enhancement stage. In the training stage, the sum of the Poisson distribution, leading to the KL divergence measure, is used as the observation model for each state of the HMM. This ensures that a computationally efficient multiplicative update can be used for the parameter update of this model. In the online enhancement stage, a novel minimum mean square error estimator is proposed for the NMF-HMM. This estimator can be implemented using parallel computing, reducing the time complexity. Moreover, compared to the traditional NMF-based speech enhancement methods, the experimental results show that our proposed algorithm improved the short-time objective intelligibility and perceptual evaluation of speech quality by 5% and 0.18, respectively.