Let $\overline{\mathcal{M}}_{g,n}$ be the Deligne-Mumford moduli space of stable algebraic curves. I would like to know which rational cohomology classes or at least which tautological classes on $\overline{\mathcal{M}}_{g,n}$ vanish when restricted to the divisor $\delta_{\mathrm{irr}}$ of curves with one non-separating node.

An example would be the top Hodge class $\lambda_g\in H^{2g}(\overline{\mathcal{M}}_{g,n})$.

I know that Arbarello and Cornalba proved here that $i^*:H^*(\overline{\mathcal{M}}_{g-1,n+2},\mathbb{Q}) \to H^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ is injective for $k\leq2g-2$ for $g\leq 7$ and for $k\leq g+5$ for $g\geq 7$.

I also understand that it's probably irrealistic to expect an explicit and general description of $\ker i^*$, either in cohomology or the tautological ring.

So here is a perhaps more approachable question:

Is $\ker i^* \subset RH^*(\overline{\mathcal{M}}_{g,n},\mathbb{Q})$ strictly bigger than the ideal generated by $\lambda_g$? I guess so! If yes, can I get some examples, maybe something working in any genus? For instance, what is there in cohomological degree between the bounds of Arbarello and Cornalba and $2g$?