Summary: The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell’s equations. Recently it has become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-\(\alpha\)) system, which differs from the original MHD system by including an additional non-linear terms (indexed by \(\alpha\)), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form \(-|\xi|^\gamma/g(|\xi|)\). In [N. Pennington, Nonlinear Anal., Real World Appl. 38, 171–183 (2017; Zbl 1379.35250)], the problem was considered with initial data in the Sobolev space \(H^{s,2}(\mathbb{R}^n)\) with \(n\geq 3\). Here we consider the problem with initial data in \(H^{s,p}(\mathbb{R}^n)\) with \(n\geq 3\) and \(p>2\). Our goal is to minimizing the regularity required for obtaining uniqueness of a solution.