Graph Neural Networks (GNNs) have played a significant role in graph representation learning. However, real-world graphs usually contain edge noise, e.g., spurious connections added to the graphs. The performance of GNNs significantly degrades when trained on such graphs. Removing the suspicious edges is a common approach to dealing with edge noise. However, current approaches ignore the impact of incomplete structure, i.e., some edges are missing. It can hinder the removal of edge noise due to the lack of sufficient edges and lead to suboptimal robust node representations. Our preliminary experiment also proves that the coexistence of edge noise and incomplete structure could further deteriorate the performance of GNNs. To this end, we propose a new framework to improve the performance of noisy graphs under incomplete structure situation. Firstly, we utilize node features to assign learnable edge weights, constructing a dense graph to alleviate incomplete structure. Subsequently, the generated edges are refined through a graph augmentation mechanism based on the estimation of edge importance, through which edge noise can be removed, and at the same time, important edges are strengthened. Finally, we propose a confidence-aware mechanism to obtain more robust node representations by fusing two views of an augmented graph. Extensive experiments on real-world graphs demonstrate that our proposed framework significantly outperforms existing methods on noisy graphs with incomplete structure.