Modern biological studies often involve testing many hypotheses organized in a group or a hierarchical structure, such as a directed acyclic graph (DAG). In these studies, researchers often wish to control the false discovery rate (FDR) after filtering the discoveries to obtain interpretable results. For addressing this goal, Katsevich, Sabatti, and Bogomolov (2023, Journal of the American Statistical Association, 118(541), 165-176) developed a general method, Focused BH, that guarantees FDR control for the filtered rejection set for a pre-specified filter, under certain assumptions. We propose improving the power of Focused BH by adapting it to group or hierarchical structures of hypotheses using data-dependent weights. The general method incorporating such weights is referred to as Weighted Focused BH (WFBH). For DAG-structured hypotheses, we propose a variant of WFBH, which can gain power by being adaptive to the DAG structure, and by exploiting the logical relationships among the hypotheses. We prove that WFBH with weights that were proposed to adapt the Benjamini-Hochberg procedure to different group structures, as well as its proposed variant for testing DAG-structured hypotheses, control the post-filtering FDR under certain assumptions. Through simulations, we demonstrate that the latter variant is robust to deviations from these assumptions and can be considerably more powerful than comparable methods. Finally, we elucidate its practical use by applying it to real datasets from microbiome and gene expression studies.