Let $\overline{\mathcal{M}}^r_{g,n}$ the space of $n$ pointed $r$stable curves of genus $g$, endowed with a root of the canonical sheaf, and let $\mathcal{C} \to \overline{\mathcal{M}}^r_{g,n}$ be the universal curve. The family $\mathcal{C}$ is endowed with a universal root of the canonical, that I denote $\mathcal{S}$. What is the relation between $\pi_!\mathcal{S}$ and the Hodge bundle $\mathbb{E}:=\pi_*\omega_{\mathcal{C}}$ on $\overline{\mathcal{M}}^r_{g,n}$? For instance what can one say about the chern classes?

A detailed computation of the Chern character of $R^\bullet \pi_\ast \mathcal S$ can be found in Alessandro Chiodo's "Towards an enumerative geometry of the moduli space of twisted curves and $r$th roots". As abx says it is a generalization of Mumford's computation of the Chern character of the Hodge bundle. Let me also say that (at least today) this is a problem that just screams for an application of GrothendieckRiemannRoch (which is indeed what Mumford and Chiodo are using). 


I assume that your $\mathcal{S}$ satisfies $\mathcal{S}^{\otimes r}\cong \omega _{\mathcal{C}/\mathcal{M}}$. Then the computation of $\pi _!\mathcal{S}$ can be done exactly as the computation of $\pi _*\omega _{\mathcal{C}/\mathcal{M}}^{\otimes n}$ in Mumford, Stability of projective varieties, L'Enseignement Mathématique, Vol. 23 (1977), §5 $$ just put $n=\frac{1}{r} $. The result is more complicated that what you suggest, it involves the boundary divisors. 

