If you're feeling overwhelmed or afraid of doing harm, it can help to have strategies in place for when you do.
If someone accuses you of bias, thank them for their honesty.
Thanking them will validate both their feelings and their choice to say something. It can be very hard for someone to come forward about bias, especially if you are in a position of authority over them. Sometimes fear and anxiety can lead people to make these statements in a way that sounds hostile or confrontational. If they know that you are open to hearing this kind of feedback and that you will not try to deny their reality, they might be less likely to maintain a hostile tone.
Above all... avoid sounding defensive, even if that's how you're feeling. Remember that one action does not define you as a person and this can be an opportunity to learn rather than a signal of failure. If you are able to in the moment, you can ask them for more information, clarification, or advice on how you could do better. Just be sure that any questions you ask indicate that you take them seriously and do not sound like you are questioning the validity of their claim.
If you don't feel ready to apologize or discuss the accusation without being defensive, tell them that you would like to reflect on what they said. This signals that you take them seriously, but that you are not going to discuss the issue at that moment.
When you are accused of bias, you should always talk with your supervisor. If the accusation comes directly from a student, it's definitely better if your supervisor hears about it from you. Being willing to discuss it signals to your supervisor that you take the impact you have on students seriously. Hopefully they will have useful advice, or be able to connect you with useful resources.
If you are upset by the situation, try to talk with someone who can help you process the experience without judgement such as a counselor, ombudsperson, etc.
Take time to reflect on what you're accused of. If it doesn't feel like an accurate assessment of your behavior, try to understand it from the other person's perspective. If you're feeling defensive, try to consider where that feeling of defensiveness is coming from.
If you feel able to re-engage in a compassionate and genuine way, give the person an opportunity to discuss it with you. It's important that you make it clear that:
You take their feelings seriously
You're open to talking
They don't owe you anything
They are free to decline to discuss it with you