Using the tight-binding description of graphene and the time dependent density matrix approach we theoretically address the nonlinear response of plasmonic graphene nanostructures to the circularly polarized light. The intensity and polarization of emitted harmonics depend on the symmetry of the system and can be analyzed applying the Neumann's principle. We nd that for the nanoakes comprising thousands of carbon atoms this is the symmetry of the carbon atom arrangement on atomic scale that determines the nonlinear response. Therefore it might be very dierent from the nonlinear response predicted using the macroscopic geometry. For the compound systems made of several nanoakes we reveal the role of the near eld interactions in intensity and circular polarization states of emitted harmonics. Finally we show that symmetry break by e.g. lattice defects strongly aects the nonlinear response of graphene nanoakes to the circularly polarized light. Our work extends theoretical studies of the nonlinear optical properties of graphene nanomaterials towards spin carrying light beams.