This paper investigates an abstract fractional Cauchy problem with damping formulated in the sense of the Caputo derivative, where the derivative orders satisfy 0<δ<γ≤1. By introducing the concept of a Caputo fractional (γ,δ,k) resolvent and systematically analyzing its fundamental properties, together with key features of the generalized Mittag–Leffler (ML) function, we establish the uniqueness and existence of strong solutions for this class of damped fractional-order evolution equations. Under more restrictive assumptions on the underlying operators, the solution admits an explicit representation in terms of ML-type functions associated with fractional exponents. Furthermore, we demonstrate that the proposed abstract framework can be effectively applied to concrete models, including fractional diffusion equations with damping. These results highlight the relevance and necessity of fractional damping models in accurately describing complex dynamical phenomena, such as vibration processes and anomalous diffusion.