When you do something petulantly, you are being irritable or peevish. If you petulantly roll your eyes at your boss, you might risk losing that promotion.
Petulantly is the adverb formation of petulant, which comes from the Latin word petulantem, meaning “wanton,” “forward,” or “insolent.” The modern use of petulantly has morphed to mean that one is moody or childish, and no longer means that one is wanton or forward. Acting petulantly won’t get you very far. People don’t usually respond well to children acting bratty, and it’s even worse when adults do it.