We investigate the effects of nonlinear absorption of the pump beam on second-harmonic generation in GaAs nanowires. Our model includes nonlinear absorption of the pump and allows obtaining a self-consistent solution of the nonlinear Maxwell equations. First, we observe that SHG conversion efficiency can be limited from two-photon absorption and generated free-carriers depending on the pump intensity. Second, we show a method to modulate the SHG response by varying the pump beam intensity. We find that varying the pump intensity from 1 GW/cm2 up to 15 GW/cm2 can red-shift the SH peak wavelength up to 5 nm and modulate the conversion efficiency at a fixed pump wavelength up to 60%. Our results enable new applications of dielectric nanoresonators for nonlinear applications such as harmonic generation, optical switching, and all-optical ultrafast modulation.