Magneto-active elastomers (MAEs) are an emerging smart composite material consisting of a compliant elastomer matrix filled with stiff, magnetizable inclusions. Meso-scale models, which treat the matrix and inclusions as discrete, are currently formulated with the assumption of perfect bonding at the interface. We apply a model for cohesive damage at the material interface and introduce a treatment for the transmission of the magnetic field in the gap. Examples demonstrate the concept and the difference in structural response between samples treated with the conventional and presented approaches.