<p>The study considers frictionless steady-state moving contact between a rigid flat or cylindrical punch and a viscoelastic half-plane coated with an exponentially graded layer. The punch is subjected to a vertical line load and moves at uniform speed along the coating surface. The coating and substrate are modeled as linear viscoelastic media governed by the Kelvin–Voigt formulation with linear hysteresis damping. Application of the Galilean transformation, Fourier integral representation, superposition, and asymptotic method yields a formulation expressed as a Cauchy-type singular integral equation of the second kind. This form arises due to the viscoelastic coupling introduced by the structure materials. The effect of the punch moving speed, gradient index, and damping factor ratio is evaluated in terms of normal contact stress and in-plane stress. Viscoelastic effects introduce asymmetry in the stress field under frictionless contact, while adjusting the gradient index of the coating mitigates peak stresses within the contact region.</p>