Boundary Expression for Chern Classes of the Hodge Bundle on Spaces of Cyclic Covers

We compute an explicit formula for the first Chern class of the Hodge Bundle over the space of admissible cyclic $\mathbb{Z}/3\mathbb{Z}$ covers of $n$-pointed rational stable curves as a linear combination of boundary strata. We then apply this formula to give a recursive formula for calculating certain Hodge integrals containing $\lambda_1$. We also consider covers with a ${\mathbb{Z}}/{2\mathbb{Z}}$ action for which we compute $\lambda_2$ as a linear combination of codimension two boundary strata.