Latent graph inference (LGI) aims to jointly learn the underlying graph structure and node representations from data features. However, existing LGI methods commonly suffer from the issue of supervision starvation, where massive edge weights are learned without semantic supervision and do not contribute to the training loss. Consequently, these supervision-starved weights, which determine the predictions of testing samples, cannot be semantically optimal, resulting in poor generalization. In this paper, we observe that this issue is actually caused by the graph sparsification operation, which severely destroys the important connections established between pivotal nodes and labeled ones. To address this, we propose to restore the corrupted affinities and replenish the missed supervision for better LGI. The key challenge then lies in identifying the critical nodes and recovering the corrupted affinities. We begin by defining the pivotal nodes as k-hop starved nodes, which can be identified based on a given adjacency matrix. Considering the high computational burden, we further present a more efficient alternative inspired by CUR matrix decomposition. Subsequently, we eliminate the starved nodes by reconstructing the destroyed connections. Extensive experiments on representative benchmarks demonstrate that reducing the starved nodes consistently improves the performance of state-of-the-art LGI methods, especially under extremely limited supervision (6.12% improvement on Pubmed with a labeling rate of only 0.3%).