Generalized inverses of tensors, such as the Moore–Penrose inverse and the Drazin inverse, have been extensively utilized in various fields including image processing and systems of multilinear algebraic equations. In this paper, the \((\mathcal {B}, \mathcal {C})\) -inverse for high-order tensors, which generalizes both the Moore–Penrose and Drazin inverses, is investigated. Firstly, the concept of one-sided regularity and the criteria for the existence of the \((\mathcal {B}, \mathcal {C})\) -inverse within the framework of high-order tensor rings under the C-product are explored. Then, algorithms for computing the strongly right \((\mathcal {B}, \mathcal {C})\) -inverse and the \((\mathcal {B}, \mathcal {C})\) -inverse of a given tensor \(\mathcal {A}\) are proposed. Finally, an application to color image deblurring is presented, demonstrating the effectiveness of the discussed generalized inverses in practical scenarios.
