  The $GW$ approximation is considered to be the simplest approximation with
Hedin's formulation of many-body perturbation theory. It is expected that some
of the deficiencies of the $GW$ approximation can be overcome by adding the
so-called vertex corrections. In this work, the recently implemented
$G_0W_0\Gamma^{(1)}_0$ scheme, which incorporates the vertex effects by adding
the full second-order self-energy correction to the $GW$ self-energy, is
applied to a set of first-row transition-metal monoxide (TMO) anions. Benchmark
calculations show that results obtained by $G_0W_0\Gamma^{(1)}_0$ on top of the
B3LYP hybrid functional starting point (SP) are in good agreement with
experiment data, giving a mean absolute error of 0.13 eV for a testset
comprising the ionization energies (IEs) of 27 outer valence molecular orbitals
(MOs) from 9 TMO anions. A systematic SP-dependence investigation by varying
the ratio of the exact exchange (EXX) component in the PBE0-type SP reveals
that, for $G_0W_0\Gamma^{(1)}_0$, the best accuracy is achieved with $20\%$
EXX. Further error analysis in terms of the orbital symmetry characteristics
(i.e, $\sigma$, $\pi$, or $\delta$) in the testset indicate the best amount of
EXX in the SP for $G_0W_0\Gamma^{(1)}_0$ calculations is independent of MO
types, and this is in contrast with the situation of $G_0W_0$ calculations
where the best EXX ratio varies for different classes of MOs. Despite its
success in describing the absolute IE values, we however found that
$G_0W_0\Gamma^{(1)}_0$ faces difficulties in describing the energy separations
between certain states of interest, worsening the already underestimated
$G_0W_0$ predictions.
