Late-breaking is almost always about news. Late-breaking news is news of events that happen just before a broadcast goes out, or just before a newspaper is printed — so late that the editors nearly missed it, and they may cut into the programme or stop the press to get it in. It is a newsroom word: urgent, a little breathless, used in headlines and presenter talk. Breaking news already means “this is happening now.” Late-breaking adds the extra pinch: it arrived at the last moment, after the running order was supposed to be finished. You would not call a late homework assignment late-breaking; keep it for news and similar last-second reports. Hyphenate it before a noun: late-breaking news, a late-breaking story.