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Many problems arisen in research fields like systems biology and signal processing can be formulated as problems of block sparse signal recovery. Generally, it is known that to pursue the sparsest solution of an underdetermined system of linear equations is non-deterministic polynomial(NP)-hard. To solve block sparse recovery problems, the algorithm of block stagewise orthogonal matching pursuit(Block-StOMP) has been proposed to recover block sparse signals from compressed measurements, which is a greedy algorithm with satisfactory practical performance and some particularly interesting theoretical properties. In this paper, we propose an improved version of Block-StOMP, termed mBlock-StOMP. Specifically, mBlock-StOMP uses the estimated TDR(true discovery rate) to prune support sets of each stage in order to decrease FAR(false alarm rate) and pursue high recovery accuracy.Moreover, rigorous theoretical analysis for mBlock-StOMP is given in this paper. Compared with Block-StOMP, simulation results demonstrate that mBlock-StOMP outperforms Block-StOMP in terms of reconstruction accuracy without increasing computational burden significantly.