Residual polymer (here, poly(methyl methacrylate), PMMA) left on graphene from transfer from metals or device fabrication processes affects its electrical and thermal properties. We have found that the amount of polymer residue left after the transfer of chemical vapor deposited (CVD) graphene varies depending on the initial concentration of the polymer solution, and this residue influences the electrical performance of graphene field-effect transistors fabricated on SiO2/Si. A PMMA solution with lower concentration gave less residue after exposure to acetone, resulting in less p-type doping in graphene and higher charge carrier mobility. The electrical properties of the weakly p-doped graphene could be further enhanced by exposure to formamide with the Dirac point at nearly zero gate voltage and a more than 50% increase of the room-temperature charge carrier mobility in air. This can be attributed to electron donation to graphene by the -NH2 functional group in formamide that is absorbed in the polymer residue. This work provides a route to enhancing the electrical properties of CVD-grown graphene even when it has a thin polymer coating.