We report on the second phase of our study of slightly rotating accretion flows onto black holes. We consider magnetohydrodynamical (MHD) accretion flows with a spherically symmetric density distribution at the outer boundary but with spherical symmetry broken by the introduction of a small, latitude-dependent angular momentum and a weak radial magnetic field. We study accretion flows by means of numerical two-dimensional, axisymmetric, MHD simulations with and without resistive heating. Our main result is that the properties of the accretion flow depend mostly on an equatorial accretion torus that is made of the material that has too much angular momentum to be accreted directly. The torus accretes, however, because of the transport of angular momentum due to the magnetorotational instability (MRI). Initially, accretion is dominated by the polar funnel, as in the hydrodynamic inviscid case, where material has zero or very low angular momentum. At the later phase of the evolution, the torus thickens toward the poles and develops a corona or an outflow or both. Consequently, the mass accretion through the funnel is stopped. The accretion of rotating gas through the torus is significantly reduced compared with the accretion of nonrotating gas (i.e., the Bondi rate). It is also much smaller than the accretion rate in the inviscid, weakly rotating case. Our results do not change if we switch on or off resistive heating. Overall our simulations are very similar to those presented by Stone, Pringle, Hawley, and Balbus despite different initial and outer boundary conditions. Thus, we confirm that MRI is very robust and controls the nature of radiatively inefficient accretion flows. Although the time-averaged properties of our models approach a steady state, we find that the instantaneous mass-accretion rate in the latter stages of our simulations is highly time-dependent, with the inner flow displaying three generic flow patterns.