Direct-fired oxy-fuel combustion as a heat source is utilized in supercritical carbon dioxide (sCO2) power cycles, such as the Allam cycle, which has shown promise in delivering higher efficiencies while achieving the complete capture of combustion products in future generation carbon-neutral power plants. The design of dedicated burners for such cycles is key in determining their overall efficiency and viability. We present a series of numerical simulations on a prototype burner, currently in development, operating in an essentially non-premixed regime with a high pressure gaseous fuel, burning in the presence of a hot oxidant flow containing a mixture of recycled CO2 and pure oxygen. Simulations rely on a turbulent combustion model based on a simplified approach for diluted steady laminar flamelets. We investigate the effect of the degree of dilution as well as pressure on the flame structure, revealing a concurrent change in stoichiometric mixture fraction and quenching scalar dissipation. We also assess the effect of injector recess and of swirl in the oxidizer stream.