Time-varying graph signal recovery has been widely used in many applications, including climate change, environmental hazard monitoring, and epidemic studies. It is crucial to choose appropriate regularizations to describe the characteristics of the underlying signals, such as the smoothness of the signal over the graph domain and the low-rank structure of the spatial-temporal signal modeled in a matrix form. As one of the most popular options, the graph Laplacian is commonly adopted in designing graph regularizations for reconstructing signals defined on a graph from partially observed data. In this work, we propose a time-varying graph signal recovery method based on the high-order Sobolev smoothness and an error-function-weighted nuclear norm regularization to enforce the low-rankness. Two efficient algorithms based on the alternating direction method of multipliers and iterative reweighting are proposed, and convergence of one algorithm is shown in detail. We conduct various numerical experiments on synthetic and real-world datasets to demonstrate the effectiveness of the proposed methods compared to the state-of-the-art in graph signal recovery.