Let M be a compact Hermitian n-manifold, and let Gamma be a symmetric convex cone. We develop a quantitative regularization method for Gamma-admissible functions. As an application, we prove Holder continuity for every bounded pluripotential solution of complex m-Hessian equations whose right-hand sides belong to L^p, for p>n/m. These results extend to a more general class of complex Hessian equations satisfying the structural condition used by Guo-Phong-Tong.