  This paper examines the general relativistic model of a geometrically thick
configuration of an accretion disc around an electrically charged black hole in
an accelerated motion, as described by the C-metric family. We aim to study the
effects of the spacetime background on the magnetised version of the thick disc
model via the sequences of figures of equilibrium. While maintaining the
assumption of non-selfgravitating (test) fluid, we newly explore the influence
of the strength of the large-scale magnetic field with field lines organised
over the length-scale of the black hole horizon. We systematically analyze the
dependence on a very broad parameter space of the adopted scenario. We
demonstrate that the C-metric can, in principle, be distinguished from Kerr
black hole metric by resolving specific (albeit rather fine) features of the
torus, such as the location of its centre, inner and outer rims, and the
overall shape. The analytical setup can serve as a testbed for numerical
simulations.
