This paper proposes an efficient mixed finite element scheme that is structure-preserving, decoupled and unconditionally stable for the magnetohydrodynamic (MHD) model. The main contributions of this paper are reflected in the following two aspects. In the MHD with vorticity formulation, the vorticity exists in both the diffusion term and the convection term, resulting in a strong coupling relationship between the velocity and vorticity. This coupling characteristic increases the difficulty of numerical solution. To address this issue, we develop a new scheme, achieving decoupling of velocity and vorticity while ensuring divergence-free property of velocity. Secondly, to further enhance computational efficiency, we utilize the scalar auxiliary variable method and ingeniously handle the coupling term, achieving decoupling of the magnetic field, current density and electric field. More importantly, this scheme can still strictly maintain the divergence-free property of the magnetic field during the decoupling process. Furthermore, our scheme not only ensures structure-preserving but also possesses the key property of unconditional energy stability. Finally, the convergence/stability tests, robustness tests and some benchmark problems (Orszag-Tang vortex and shear layer) of the scheme were verified by numerical experiments.