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We compute an explicit formula for the first Chern class of the Hodge Bundleover 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 certainHodge integrals containing $\lambda_1$. We also consider covers with a${\mathbb{Z}}/{2\mathbb{Z}}$ action for which we compute $\lambda_2$ as alinear combination of codimension two boundary strata.

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