  We consider the low-temperature expansion of the Casimir-Polder free energy
for an atom and graphene by using the Poisson representation of the free
energy. We extend our previous analysis on the different relations between
chemical potential $\mu$ and mass gap parameter $m$. The key role plays the
dependence of graphene conductivities on the $\mu$ and $m$. For simplicity, we
made the manifest calculations for zero values of the Fermi velocity. For $\mu
>m$ the thermal correction $\sim T^2$ and for $\mu < m$ we confirm the recent
result of Klimchitskaya and Mostepanenko, that the thermal correction $\sim
T^5$. In the case of exact equality $\mu =m$ the correction $\sim T$. This
point is unstable and the system falls to the regime with $\mu >m$ or $\mu <m$.
The analytical calculations are illustrated by numerical evaluations for the
Hydrogen atom/graphene system.
