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In this paper, semi-supervised learning and clustering algorithms on hypergraph cutting are studied; Firstly, by discussing hypergraph cutting and hyperedge expansion methods as well as its cutting function, the total variation on hypergraph is introduced as a Lovasz extension of hypergraph cutting. Based on this, a set of regularization functions are proposed, which correspond to Laplacian regularization on graph; Then, a semi-supervised learning method based on regularization function family is proposed, and a spectral clustering method based on balanced hypergraph cutting is proposed; In order to solve these two learning problems, they are transformed into solving convex optimization problems, and a scalable algorithm whose main component is proximal mapping is proposed to realize semi-supervised learning and clustering; Simulation results show that the proposed semi-supervised learning and clustering methods based on hypergraph cutting has better standard deviation and clustering error performance than the classical hyperedge expansion and other graph cutting methods.