We apply a circularly and linearly polarized terahertz field on a monolayer of graphene, taking into account spin-orbit interactions of the intrinsic and Rashba types. It turns out that the field can be used not only to induce a gap in the energy spectrum, but also to close an existing gap due to the different reaction of the spin components with circularly polarized light. Signatures of spin-orbit coupling in the density of states of the driven system can be observed even for energies where the static density of states is independent of spin-orbit interactions. Furthermore it is shown that the time evolution of the spin polarization and the orbital dynamics of an initial wave packet can be modulated by varying the ratio of the spin-orbit-coupling parameters. Assuming that the system acquires a quasistationary state, the optical conductivity of the irradiated sample is calculated. Our results confirm the multistep nature of the conductivity obtained recently, where the number of intermediate steps can be changed by adjusting the spin-orbit-coupling parameters and the orientation of the field.