  This paper investigates stationary mean-field games (MFGs) on the torus with
Lipschitz non-homogeneous diffusion and logarithmic-like couplings. The primary
objective is to understand the existence of $C^{1,\alpha}$ solutions to address
the research gap between low-regularity results for bounded and measurable
diffusions and the smooth results modeled by the Laplacian.
  We use the Hopf--Cole transformation to convert the MFG system into a scalar
elliptic equation. Then, we apply Morrey space methods to establish the
existence and regularity of solutions. The introduction of Morrey space methods
offers a novel approach to address regularity issues in the context of MFGs.
