  The rigorous finite-temperature QED formalism of the polarization tensor is
used to study the combined effect of nonzero mass gap $m$ and chemical
potential $\mu$ on the Casimir force and its thermal correction in the
experimentally relevant configuration of a Au sphere interacting with a real
graphene sheet or with graphene-coated dielectric substrates made of different
materials. It is shown that for both a free-standing graphene sheet and for
graphene-coated substrates the magnitude of the Casimir force decreases as $m$
is increased, while it increases as $\mu$ is increased, indicating that these
parameters act in opposite directions. According to our results, the impact of
$m$ and/or $\mu$ on the Casimir force for graphene-coated plates is much
smaller than for a free-standing graphene sheet. Furthermore, computations show
that the Casimir force is much stronger for graphene-coated substrates than for
a free-standing graphene sample, but the thermal correction and its fractional
weight in the total force are smaller in the former case. These results are
applied to a differential setup that was recently proposed to observe the giant
thermal effect in the Casimir force for graphene. We show that this experiment
remains feasible even after taking into account the influence of the nonzero
mass-gap and chemical potential of real graphene samples. Possible further
applications of the obtained results are discussed.
