We study a general system defined by a second order evolution equation, coupled with a first order differential equation, which can model some classes of dynamic thermal contact problems. We present and establish an existence and uniqueness result, by using general results on evolution equations and Banach's contraction principle, with monotone operators and fixed point arguments. Then a fully discrete scheme for numerical approximations and analysis of error order estimate are provided. Finally applications to concrete dynamic contact problems are given, followed by numerical simulations.