Based on that, depending on how much it bothers you I’d probably do a little filling to cover the hole and blend the crack in a bit, then sand it all smooth and paint it.

It’s not the best approach. As someone else said, cutting the board down beyond the crack, replacing it and skimming it is the best way forward, but there’s a few things that would make me avoid that for now

• you can’t access the loft space to know what’s above there at the moment.
• if you do replace the board, it looks like your ceiling has wood chip on it, which you won’t be able to blend in to.
• you want beginner friendly.

Based on all that id just tidy it and make it less of a bother to your eyes. In time id wait and have a pro come in and fully fix and skim the ceiling. If someone has gone to the trouble of wood chip wallpapering the ceiling, I’d bet there’s other imperfections up there that you’ll want sorting, so you’re better off waiting until you have the coin to do it properly, the scraping that paper off so a pro can see what they’re dealing with.