  We study the effect of a dilute homogeneous spatial distribution of
non-magnetic impurities on the equilibrium supercurrent sustained by a
ballistic graphene Josephson junction in the short junction limit. Within the
Dirac-Bogoliubov-de Gennes approach and modeling impurities by the Anderson
model we derive the supercurrent and its equilibrium power spectrum. We find a
modification of the current-phase relation with a reduction of the skewness
induced by disorder, and a nonmonotonic temperature dependence of the critical
current. The potentialities of the supercurrent power spectrum for accurate
spectroscopy of the hybridized Andreev bound states-impurities spectrum are
highlighted. In the low temperature limit, the supercurrent zero frequency
thermal noise directly probes the spectral function at the Fermi energy.
