Bemusedly is the adverb: doing something in a puzzled way. It tells you how an action was done — "he shook his head bemusedly," "she smiled bemusedly at the menu." It goes with small physical actions that show a face or body reacting: smiling, blinking, staring, shaking the head, watching. It is a rare and rather literary word. You will find it in novels; you will almost never hear it spoken. In real speech, people go around it — they say "he looked bemused" or "he seemed a bit confused" instead. So it is worth being able to read, and safe to leave out of your own writing until you feel sure of it.