  Motivated by recent experimental progress in the realization of hybrid
structures with a topologically superconducting nanowire coupled to a quantum
dot, viewed through the lens of the emerging field of correlated Majorana
fermions, we introduce a class of interacting Majorana-Anderson impurity models
which admit an exact solution for a wide range of parameters, including on-site
repulsive interactions of arbitrary strength. The model is solved by mapping it
via the $\mathbb{Z}_2$ slave-spin method to a noninteracting resonant level
model for auxiliary Majorana degrees of freedom. The resulting gauge constraint
is eliminated by exploiting the transformation properties of the Hamiltonian
under a special local particle-hole transformation. For a spin-polarized Kitaev
chain coupled to a quantum dot, we obtain exact expressions for the dot
spectral functions at both zero and finite temperature. We study how the
interaction strength and localization length of the end Majorana zero mode
affect physical properties of the dot, such as quasiparticle weight, double
occupancy, and odd-frequency pairing correlations, as well as the local
electronic density of states in the superconducting chain.
