  We present and benchmark a self-energy approach for quasiparticle energy
calculations that goes beyond Hedin's $GW$ approximation by adding the full
second-order self-energy (FSOS-$W$) contribution. The FSOS-$W$ diagram involves
two screened Coulomb interaction ($W$) lines and adding the FSOS-$W$ to the
$GW$ self-energy can be interpreted as first-order vertex correction to $GW$
($GW\Gamma^{(1)}$). Our FSOS-$W$ implementation is based on the
resolution-of-identity technique and exhibits better than $O(N^5)$ scaling with
system size for small to medium-sized molecules. We then present one-shot
$GW\Gamma^{(1)}$ ($G_0W_0\Gamma^{(1)}_0$) benchmarks for the $GW$100 test set
and a set of 24 acceptor molecules. For semilocal or hybrid density functional
theory starting points, $G_0W_0\Gamma^{(1)}_0$ systematically outperforms
$G_0W_0$ for the first vertical ionization potentials (vIPs) and electron
affinities (vEAs) of both test sets. Finally, we demonstrate that a static
FSOS-$W$ self-energy significantly underestimates the quasiparticle energies.
