We study a general system defined by a second order evolution inequality, coupled with a first order differential equation, which can model some classes of dynamic thermal contact problems with friction. We present and establish an existence and uniqueness result, by using general results on first order evolution inequality, with monotone operators and fixed point methods. Then a fully discrete scheme for numerical approximations and analysis of error order estimate are provided. Finally application to concrete thermal dynamic contact problems will be given, completed by numerical simulations.