<p>In this paper, we study the degenerate or singular fully nonlinear dead-core systems coupled with strong absorption terms. We establish several properties, including improved regularity of viscosity solutions along the free boundary, non-degeneracy, a measure estimate of the free boundary, Liouville-type results, and the behavior of blow-up solution. We also derive sharp regularity estimates for viscosity solutions to Hénon-type equations with a degenerate weight and strong absorption, governed by a degenerate fully nonlinear operator. Our results are new even for the model equations involving degenerate Laplacian operators.</p>