  The Casimir-Polder force between atoms or nanoparticles and graphene-coated
dielectric substrates is investigated in the region of large separations.
Graphene coating with any value of the energy gap and chemical potential is
described in the framework of the Dirac model using the formalism of the
polarization tensor. It is shown that the Casimir-Polder force from a
graphene-coated substrate reaches the limit of large separations at
approximately 5.6 $\mu$m distance between an atom or a nanoparticle and
graphene coating independently of the values of the energy gap and chemical
potential. According to our results, however, the classical limit, where the
Casimir-Polder force no longer depends on the Planck constant and the speed of
light, may be attained at much larger separations depending on the values of
the energy gap and chemical potential. In addition, we have found a simple
analytic expression for the Casimir-Polder force from a graphene-coated
substrate at large separations and determined the region of its applicability.
It is demonstrated that the asymptotic results for the large-separation
Casimir-Polder force from a graphene-coated substrate are in better agreement
with the results of numerical computations for the graphene sheets with larger
chemical potential and smaller energy gap. Possible applications of the
obtained results in nanotechnology and bioelectronics are discussed.
