In this paper, we introduce a new generalized inverse, called MPWG inverse of a complex square matrix. We investigate characterizations, representations, and properties for this new inverse. Then, by using the core-EP decomposition, we discuss the relationships between MPWG inverse and other generalized inverses. A variant of the successive matrix squaring computational iterative scheme is given for calculating the MPWG inverse. The Cramer rule for the solution of a singular equation Ax = b is also presented. Moreover, the MPWG inverse being used in solving appropriate systems of linear equations is established. Finally, we analyze the MPWG binary relation.