We propose a conditional gradient framework for a composite convex minimization template with broad applications. Our approach combines the notions of smoothing and homotopy under the CGM framework, and provably achieves the optimal O(1/√k) convergence rate. We demonstrate that the same rate holds if the linear subproblems are solved approximately with additive or multiplicative error. Specific applications of the framework include the non-smooth minimization, semidefinite programming, and minimization with linear inclusion constraints over a compact domain. We provide numerical evidence to demonstrate the benefits of the new framework.