Generating text from abstract meaning repre-sentation (AMR) is a challenging task. Graph-to-sequence (Graph2Seq-based) methods and pre-trained-based methods are proposed for this task. However, both methods have advantages and disadvantages. Graph2Seq-based methods can make use of the structural information of the graph but not the extra knowledge, while pre-trained-based methods have the advantage of the utilization of extra knowledge but may lose the structural information. In addition, both types of methods often suffer from the under- and over-translation problem. To address these prob-lems, we propose a graph structure reconstruction and coverage enhanced model for this task. The graph structure reconstruction uses two auxiliary objectives, relationship prediction and distance prediction of nodes in AMR graphs to enhance the information of graph structure. In addition, we design a coverage mechanism to solve the problem of information under-translation or over-translation in AMR-to-text generation. Experimental results on three datasets show that our proposed method outperforms the existing methods significantly.