I'm not sure if she's trying to make a joke or pun, but it does look like she's being punished for being "cute".

As always, puns are hard to translate cleanly, but basically she's just saying her name (子日, Nenohi), which happens to mean New Year's Day, as an answer to her own question of "what day is it?". It's a really, really stupid joke (if you can even call it that, it's more like an excuse to shout her name), and so the fleet carriers aren't taking any nonsense from the destroyer.

Side note: Just acquired her in the game today, and she makes a comment about her name being difficult to read (as the の from 子の日 gets dropped when it's a name)

"今日は何の日？子日だよぉー! (What day is today? Day of The Rat!)" is one of Nenohi's quotes when you click on her as a secretary.