We consider super-solutions to fully nonlinear elliptic equations in the presence of measurable ingredients. Our analysis explores the consequences of the weak Harnack inequality, combined with one-sided geometric control, typically related to a generalised maximum principle. We prove regularity estimates for semi-convex \( L^p \)-viscosity super-solutions in the Escauriaza regime, both in Hölder and Lipschitz-continuous spaces. We also consider super-solutions satisfying a Hölder-type modulus of continuity from below and establish local Hölder-regularity, both at the level of the function and at the level of the gradient. As a consequence, we study the quality of the diffusion associated with viscosity super-solutions; it stems from the connection of such functions with the fractional Laplacian. We close the paper with a consequence of the weak Harnack inequality concerning the class of viscosity solutions.