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An analytical model is presented for the analysis of multilayer wire media loaded with 2-D arrays of thin material terminations, characterized in general by a complex surface conductivity. This includes the cases of resistive, thin metal, or graphene patches and impedance ground planes. The model is based on the nonlocal homogenization of the wire media with additional boundary conditions (ABCs) at the connection of thin (resistive) material. Based on charge conservation, new ABCs are derived for the interface of two uniaxial wire mediums with thin imperfect conductors at the junction. To illustrate the application of the analytical model and to validate the new ABCs, we characterize the reflection properties of multilayer absorbing structures. It is shown that in such configurations the presence of vias results in the enhancement of the absorption bandwidth and an improvement in the absorptivity performance for increasing angles of an obliquely incident TM-polarized plane wave. The results obtained using the analytical model are validated against full-wave numerical simulations.