@FrancescoPolizzi : You are correct that it only has quotient singularities. Does this imply that $H^k$ is pure of weight $k$? I can't find a statement like that in any of the surveys I've consulted, but as I said I am not an expert in this area.

@Ari : So the whole theory of weights works for stacks? Is there a down-to-earth reference for that? I was reading the paper just thinking of the coarse moduli spaces. Is there a way to see this using from the orbifold perspective, i.e. from the fact that there is a finite orbifold cover that is smooth (such a thing was constructed by de Jong-Pikaart).?

@Ari : Yes, but $\overline{\mathcal{M}}_{g,p}$ is not smooth; it has singularities along its boundary.