As a type of main optical error source in the resonant micro-optic gyro (RMOG), the optical Kerr-effect brings a nonzero bias to the output of the RMOG. The light-intensity fluctuations are interpreted as the cause. To eliminate the drifts due to the optical Kerr-effect, the intensities of the clockwise (CW) and counterclockwise (CCW) lightwaves circulating in the resonator should be equal at all times. Through theoretical analysis and experimental investigation, a linear relationship between the second-harmonic demodulated signal and the light-intensity input to the resonator is demonstrated for the sinusoidally phase modulated RMOG. Both our numerical simulation and experimental verification are carried out, which, for the first time to the best of our knowledge, demonstrate that the second-harmonic demodulated signal can be used as a feedback error signal to reduce both the input-intensity mismatch between the CW and CCW lightwaves and their intensity fluctuations. By applying the light-intensity feedback loop to the closed-loop RMOG, the light-intensity fluctuations are reduced to 2.7×10⁻⁵, down from 5.86%. As a result, the optical-Kerr effect induced error is effectively reduced.