  The Casimir-Polder force between spherical nanoparticles and a
graphene-coated silica plate is investigated in situations out of thermal
equilibrium, i.e., with broken time-reversal symmetry. The response of graphene
coating to the electromagnetic field is described on the basis of first
principles of quantum electrodynamics at nonzero temperature using the
formalism of the polarization tensor in the framework of the Dirac model. The
nonequilibrium Casimir-Polder force is calculated as a function of the mass-gap
parameter, chemical potential of graphene and temperature of the
graphene-coated plate, which can be both higher and lower than that of the
environment. It is shown that the force value increases with increasing
chemical potential, and this increase is more pronounced when the temperature
of a graphene-coated plate is lower than that of the environment. The
nonequilibrium force also increases with increasing temperature of the
graphene-coated plate. This increase is larger when the plate is hotter than
the environment. The effect is revealed that the combined impact of the
chemical potential $\mu$ and mass gap $\Delta$ of graphene coating depends on
the relationship between $\Delta$ and 2$\mu$. If $2\mu>\Delta$ the magnitude of
the nonequilibrium force between nanoparticles and a cooled graphene-coated
plate becomes much larger than for a graphene coating with $\mu=0$. The
physical reasons explaining this effect are elucidated. Possible applications
of the obtained results are discussed.
