  The full quark-gluon vertex is a crucial ingredient for the dynamical
generation of a constituent quark mass from the standard quark gap equation,
and its non-transverse part may be determined exactly from the nonlinear
Slavnov-Taylor identity that it satisfies. The resulting expression involves
not only the quark propagator, but also the ghost dressing function and the
quark-ghost kernel, and constitutes the non-abelian extension of the so-called
"Ball-Chiu vertex", known from QED. In the present work we carry out a detailed
study of the impact of this vertex on the gap equation and the quark masses
generated from it, putting particular emphasis on the contributions directly
related with the ghost sector of the theory, and especially the quark-ghost
kernel. In particular, we set up and solve the coupled system of six equations
that determine the four form factors of the latter kernel and the two typical
Dirac structures composing the quark propagator. Due to the incomplete
implementation of the multiplicative renormalizability at the level of the gap
equation, the correct anomalous dimension of the quark mass is recovered
through the inclusion of a certain function, whose ultraviolet behavior is
fixed, but its infrared completion is unknown; three particular Ans\"atze for
this function are considered, and their effect on the quark mass and the pion
decay constant is explored. The main results of this study indicate that the
numerical impact of the quark-ghost kernel is considerable; the transition from
a tree-level kernel to the one computed here leads to a $20\%$ increase in the
value of the quark mass at the origin. Particularly interesting is the
contribution of the fourth Ball-Chiu form factor, which, contrary to the
abelian case, is nonvanishing, and accounts for $10\%$ of the total constituent
quark mass.
