Graph neural networks (GNNs) have achieved remarkable results for various graph learning tasks. However, one of the recent challenges for GNNs is to adapt to different types of graph inputs, such as heterophilic graph datasets in which linked nodes are more likely to contain a different class of labels and features. Accordingly, an ideal GNN model should adaptively accommodate all types of graph datasets with different labeling distributions. In this paper, we tackle this challenge by proposing a regularization framework on graph framelet with the regularizer induced from graph p-Laplacian. By adjusting the value of p, the p-Laplacian based regularizer restricts the solution space of graph framelet into the desirable region based on the graph homophilic features. We propose an algorithm to effectively solve a more generalized regularization problem and prove that the algorithm imposes a (p-Laplacian based) spectral convolution and diagonal scaling operation to the framelet filtered node features. Furthermore, we analyze the denoising power of the proposed model and compare it with the predefined framelet denoising regularizer. Finally, we conduct empirical studies to show the prediction power of the proposed model in both homophily undirect and heterophily direct graphs with and without noises. Our proposed model shows significant improvements compared to multiple baselines, and this suggests the effectiveness of combining graph framelet and p-Laplacian.