The cold start problem in recommender systems remains a critical challenge. Current solutions often train hybrid models on auxiliary data for both cold and warm users/items, potentially degrading the experience for the latter. This drawback limits their viability in practical scenarios where the satisfaction of existing warm users/items is paramount. Although graph neural networks (GNNs) excel at warm recommendations by effective collaborative signal modeling, they haven't been effectively leveraged for the cold-start issue within a user-item graph, which is largely due to the lack of initial connections for cold user/item entities. Addressing this requires a GNN adept at cold-start recommendations without sacrificing performance for existing ones. To this end, we introduce Graph Neural Patching for Cold-Start Recommendations (GNP), a customized GNN framework with dual functionalities: GWarmer for modeling collaborative signal on existing warm users/items and Patching Networks for simulating and enhancing GWarmer's performance on cold-start recommendations. Extensive experiments on three benchmark datasets confirm GNP's superiority in recommending both warm and cold users/items.