hi, could anyone give me a recap of what happens during the moon (you beat up kaguya but I don't remember why) and what happens up until there's no more story?

and also if the Hillary npc does anything significant?

Hillary remains useless, much like the real life version.

You beat up Kaguya because she gets possessed by a dress, and currently we're trying to stop a boat from being used as a weapon against Earth. May have something to do with Susanoo and Amaterasu's brother, Tsukuyomi, who got name-dropped a lot.