The cumulant expansion of the Green's function is a computationally efficient beyond-$GW$ approach renowned for its significant enhancement of satellite features in materials. In contrast to the ubiquitous $GW$ approximation of many-body perturbation theory, \textit{ab initio} cumulant expansions performed on top of $GW$ ($GW$+C) have demonstrated the capability to handle multi-particle processes by incorporating higher-order correlation effects or vertex corrections, yielding better agreements between experiment and theory for satellite structures. While widely employed in condensed matter physics, very few applications of $GW$+C have been published on molecular systems. Here, we assess the performance of this scheme on a series of 10-electron molecular systems (\ce{Ne}, \ce{HF}, \ce{H2O}, \ce{NH3}, and \ce{CH4}) where full configuration interaction estimates of the outer-valence quasiparticle and satellite energies are available.