Based on the m-core-nilpotent decomposition, we introduce a generalized inverse named the m-MP weak core inverse, which unifies the CMP inverse and the MP weak core inverse. Some properties, characterizations and representations of this generalized inverse are shown. Then, the relationship between the m-MP weak core inverse and a nonsingular bordered matrix is established. A variant of the successive squaring computational iterative scheme is given for calculating the m-MP weak core inverse. An equivalent condition for the continuity for the m-MP weak core inverse is also studied. In the final, the m-MP weak core inverse is used in solving a system of linear equations.
Tan et al. (2025) studied this question.