Employing the Cooper-Rushton-Grimes (CRG) interatomic empirical potential with new parameters for U3+ interactions in U1-y,PuyO2-x, classical Molecular Dynamics simulations were carried out for the first time to compute the enthalpy, the lattice parameter, the elastic constants and derived properties of hypostoichiometric U1-y,PuyO2-x compounds (with y in the range 0-1, x between 1.92-2.0, and from 1000 K to melting points). As hypostoichiometry results from adding oxygen vacancies to U1-y,PuyO2, this study investigates how oxygen vacancies affect the thermodynamic and thermomechanical properties of MOX compounds. The most pronounced effect appears at high temperatures, where the so-called Bredig transition occurs. We observed that the effects of this transition on the properties are softened by the presence of oxygen vacancies. The analytical law of the heat capacity for stoichiometric mixed-oxide fuels U1-y,PuyO2 developed by Bathellier et al. [9] was modified by introducing the deviation from stoichiometry as a variable. It was fitted on Molecular Dynamics data of heat capacity and linear thermal expansion coefficient and a good agreement was found. To model the thermal evolution of mechanical properties up to the melting point, a new law was developed. The laws may be used in Fuel Performance Codes.