Speaker verification is hampered by background noise, particularly at extremely low Signal-to-Noise Ratio (SNR) under 0 dB. It is difficult to suppress noise without introducing unwanted artifacts, which adversely affects speaker verification. We proposed the mechanism called Gradient Weighting (Grad-W), which dynamically identifies and reduces artifact noise during prediction. The mechanism is based on the property that the gradient indicates which parts of the input the model is paying attention to. Specifically, when the speaker network focuses on a region in the denoised utterance but not on the clean counterpart, we consider it artifact noise and assign higher weights for this region during optimization of enhancement. We validate it by training an enhancement model and testing the enhanced utterance on speaker verification. The experimental results show that our approach effectively reduces artifact noise, improving speaker verification across various SNR levels.