Incomplete multi-view clustering aims to exploit the information of multiple incomplete views to partition data into their clusters. Existing methods only utilize the pair-wise sample correlation and pair-wise view correlation to improve the clustering performance but neglect the high-order correlation of samples and that of views. To address this issue, we propose a high-order correlation preserved incomplete multi-view subspace clustering (HCP-IMSC) method which effectively recovers the missing views of samples and the subspace structure of incomplete multi-view data. Specifically, multiple affinity matrices constructed from the incomplete multi-view data are treated as a third-order low rank tensor with a tensor factorization regularization which preserves the high-order view correlation and sample correlation. Then, a unified affinity matrix can be obtained by fusing the view-specific affinity matrices in a self-weighted manner. A hypergraph is further constructed from the unified affinity matrix to preserve the high-order geometrical structure of the data with incomplete views. Then, the samples with missing views are restricted to be reconstructed by their neighbor samples under the hypergraph-induced hyper-Laplacian regularization. Furthermore, the learning of view-specific affinity matrices as well as the unified one, tensor factorization, and hyper-Laplacian regularization are integrated into a unified optimization framework. An iterative algorithm is designed to solve the resultant model. Experimental results on various benchmark datasets indicate the superiority of the proposed method. The code is implemented by using MATLAB R2018a and MindSpore library: https://github.com/ChangTang/HCP-IMSC.