Can’t really tell from the picture if it’s mould or not but it would be fairly sensible to assume it was.
I’m guessing that if it’s a bit of beekeeping equipment It was fairly expensive and you probably want to make it last as long as possible.
The outside of the box:
I would sand it down, put some nice weather proofing on the outside. There are loads of exterior wood paints in most diy shops. There’s even really expensive bee friendly paints you can buy if budget is not a concern - but as the paint is on the outside I’m not convinced yet it’s worth the extra dosh.
The inside:
Maybe sand down if it’s on the inside too, or maybe just scorch it. Probably scorch it too for good measure. Don’t paint, treat or weather proof the inside of the box