We consider a Cahn-Hilliard-Boussinesq system with positive heat diffusivity and singular potential on a two-dimensional bounded domain with suitable boundary conditions. For the corresponding initial and boundary value problem we prove the existence of strong solutions and the well-posedness for weak solutions. Then we set the diffusivity equal to zero. In this case, the model can be viewed as an approximation of the two-dimensional compressible Navier-Stokes-Cahn-Hilliard system proposed in [

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