We will discuss the regularity of viscosity solutions for fully nonlinear Hessian equations with coefficients in some Muckenhoupt class. We prove Holder and higher regularity under mild assumptions on the coefficients. We use approximation techniques in the spirit of Caffarelli's seminal approach. A special attention has to be paid to the degeneracy of the coefficients and we assume some natural structural assumptions which entail applications to curvature equations on conic manifolds.