Annoyingly means “in a way that makes you a little angry, or tired of something.” The feeling is small and sharp, not huge rage. You can put it before an adjective: annoyingly loud music, an annoyingly slow computer. Very often English puts it at the front of a sentence: “Annoyingly, the shop had just closed.” That means “this fact is annoying.” Younger speakers use that sentence shape a lot. The word is not rude by itself, but it does complain. Though actually, if something is cruel or truly dangerous, annoyingly is too small a word — you would not use it for a disaster. It is the adverb of annoying, and annoying is already milder than furious.