When you do something cagily, you do it in a closed, careful way. You do not want other people to see what you think, what you plan, or how much you know. It is the adverb of cagey (also spelled cagy). People use it about talk more than about moving your body: you answer cagily, you smile cagily, you watch someone cagily. In sport writing you will also see a team play cagily — they will not take a risk, they will not show their plan. It is a little informal, and it is not a hard, angry word; it is the word for someone who keeps their cards close. Though actually, it is not the same as carefully. Carefully can mean you do not want to break a cup. Cagily means you do not want to give yourself away.