  In this paper we introduce and study the weak Gorenstein global dimension of
a ring $R$ with respect to a left $R$-module $C$. We provide several
characterizations of when this homological invariant is bounded. Two main
applications are given: first, we prove that the weak Gorenstein global
dimension of $R$ relative to a semidualizing $(R,S)$-bimodule $C$ can be
computed either by the ${\rm G_C}$-flat dimension of the left $R$-modules or
right $S$-modules, just like the (absolute) weak global dimension. As a
consequence, a new argument for solving Bennis' conjecture is obtained. As a
second application, we give a concrete description of the weak equivalences in
the ${\rm G_C}$-flat model structure recently found by the authors. In order to
prove this result, an interesting connection between abelian model structures
and AB-weak contexts is proved. This connection leads to a result that can be
applied to obtain abelian model structures with a simpler description of
trivial objects.
