Beguilingly means in a beguiling way: someone smiles, speaks, or moves so that you are pulled in, and you may stop thinking clearly. It is an adverb, so it tells you how a thing is done, and it sits next to a verb: smiled beguilingly, spoke beguilingly. The feeling is the same as with beguiling — pleasant, a little dangerous, not a simple kind word. You meet it in stories and careful writing more than in speech. It has nothing to do with begging.