The phrase "bit at" is typically used to describe a situation where someone is persistently bothering or annoying someone else. There are a number of synonyms that can be used to express the same idea, including "pester," "harass," "irritate," "annoy," and "bother." Other synonyms include "provoke," "aggravate," "vex," "nag," and "badger." All of these words convey the idea of continued bothering or pestering, and can be used in a variety of contexts, whether you're talking about a child who won't stop asking questions, a pestering salesperson, or a partner who keeps nagging you about household chores.