  Recently, Malik and Ferreyra introduced the $m$-weak core inverse for complex
square matrices which generalizes the core-EP inverse, the WC inverse, and
therefore the core inverse. The main aim of this paper is to extend the concept
of $m$-weak core inverse for complex rectangular matrices. This extension is
called the $W$-weighted $m$-weak core inverse. We analyse its existence and
uniqueness as solution of a system of matrix equations. We present various
properties, representations and characterizations of the $W$-weighted $m$-weak
core inverse, as well as its applications in solving certain matrix systems. A
canonical form of the $W$-weighted $m$-weak core inverse is also provided by
using a simultaneous unitary block upper triangularization of a pair of
rectangular matrices.
