The study of magnetohydrodynamics (MHD) is fundamental to understanding the dynamics of electrically conducting fluids, such as plasmas and liquid metals. Classical MHD models, however, fail to capture phenomena at small scales where electron and ion motions decouple. The Hall--MHD equations extend classical MHD by incorporating the Hall effect, which becomes significant at small length scales. This introduces a dispersive, strongly nonlinear term in the induction equation, leading to richer dynamics, including whistler waves and fast magnetic reconnection corresponding to physically observable phenomena in astrophysics. Several numerical approaches have been proposed for the Hall–MHD system. These include a linearly implicit spectral scheme and a structure-preserving finite element method for the stationary problem. However, neither study provides a rigorous error analysis. We propose a linear, energy-stable, divergence-preserving mixed finite element method for the numerical discretisation of the unsteady Hall--MHD equations. Our approach proceeds in three main steps. First, to improve numerical stability and well-posedness, we introduce a Voigt-type regularisation of the Hall--MHD system, which may be interpreted as a simplified representation of electron inertia and finite Larmor radius effects in the underlying plasma model. Second, we formulate the problem in a mixed setting based on compatible finite element spaces, ensuring consistency with the intrinsic energy law and divergence-free constraints of the system. Finally, we employ a fully discrete first-order time discretisation together with a structure-preserving linearisation of the nonlinear terms, exploiting intrinsic cancellations to obtain a linear scheme that retains the key invariants of the continuous model. Exploiting the properties induced by the compatible finite element spaces, we establish a rigorous error analysis proving optimal-order convergence under appropriate regularity assumptions on the solution. Numerical experiments corroborate the theoretical findings. Extensions to stochastically perturbed Hall--MHD system will be briefly discussed, if time permits.