Can I avoid my mom's boyfriend?
Why would you avoid him?
I hear he is badmouthing her behind her back
That's awful and that would make me want to avoid him, too. I would try to verify the accusations and if you believe they are true then speak to your mother about this information. She deserves a partner who treats her with care and respect.
Nah. It doesn't matter because I can't trust him.
You not trusting him is even more reason to tell your mom the truth. Do you want her to continue to be in a toxic relationship with someone you can't trust? She might be better off alone or with someone that actually cares about her.