We investigate self-consistent, steady-state axisymmetric solutions of an incompressible tokamak plasma using a visco-resistive magnetohydrodynamic model. A key contribution of this work is the formulation of Poisson’s equation that governs the pressure profile. Our analysis reveals that the current modelling fails to produce realistic pressure levels. To overcome this limitation, we introduce additional non-inductive current drives, akin to those generated by neutral beam injection or radio frequency heating, modelled as modifications to the toroidal current. Numerical simulations validate our enhanced model, showing significant improvements in pressure profile characteristics. In the cases examined, the effect of these current drives on the velocity profiles is moderate, except when the non-inductive current drives induce reversals in the total toroidal current density, leading to non-nested flux surfaces with internal separatrices.