Weight 2 cohomology of graph complexes of cyclic operads and the handlebody group

We compute the weight 2 cohomology of the Feynman transforms of the cyclic (co)operads $\mathsf{B}\mathsf{V}$ and $\mathsf{H}\mathsf{y}\mathsf{C}\mathsf{o}\mathsf{m}$, and the top$-2$ weight cohomology of the Feynman transforms of $D\mathsf{B}\mathsf{V}$ and $\mathsf{G}\mathsf{r}\mathsf{a}\mathsf{v}$. Using a result of Giansiracusa, we compute, in particular, the top$-2$ weight cohomology of the handlebody group. We compare the result to the top$-2$ weight cohomology of the moduli space of curves ${\mathcal{M}}_{g,n}$, recently computed by Payne and the last-named author. We also provide another proof of a recent result of Hainaut-Petersen identifying the top weight cohomology of the handlebody group with the Kontsevich graph cohomology.