In this paper, we introduce the m-WG inverse in Minkowski space. Firstly, we show the existence and the uniqueness of the m-WG inverse. Secondly, we give representations of the m-WG inverse. Thirdly, we characterize the m-WG inverse by applying a bordered matrix. In addition, we extend the generalized Cayley-Hamilton theorem to the m-WG inverse matrix. Finally, we apply the m-WG inverse to solve linear equations in Minkowski space