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In this paper, we investigate a fully discrete finite element scheme for the incompressible magnetohydrodynamic (MHD) equations with magnetic-current formulation that was introduced in Hu et al. (2016). We discretize the system by the semi-implicit Euler scheme in time and a mixed finite element approach together with finite element exterior calculus in space. The resulting scheme enjoys the structure-preserving feature that it can always produce an exactly divergence-free magnetic induction on the discrete level. The unique solvability and unconditional stability of the scheme are also proved rigorously. By utilizing the energy argument, error estimates for the velocity, magnetic induction, current density and induced electric field are further established under the low regularity hypothesis for the exact solutions. Numerical results are provided to verify the theoretical analysis and to show the effectiveness of the proposed scheme. • Error analysis of a fully discrete finite element method for the MHD equations is given. • The proposed scheme is linear, unconditional stability and structure-preserving. • Numerical examples are provided to verify the theoretical findings. [ABSTRACT FROM AUTHOR]