Aibu to just go in and give the place a good scrub.
Yes, you are, without asking her. We all have different standards and what’s ‘grubby’ for some people is completely normal for someone else. There’s nothing worse than someone coming in and cleaning and tidying your house without asking. It’s really disrespectful.
Working full time as a lone parent is very hard work. There’s never enough hours in the day and it’s important to prioritise time with your child over housework. There’s no one to get you a drink when you finally sit down when the kids are in bed. Or encourage you to keep going and get stuck into jobs together.
We all parent differently. My DSs had longish hair when they were young. As long as it’s not in his eyes and annoying him, and he’s happy with it, it doesn’t matter. Kids that age are forever growing out of clothes and getting holes in them etc.
As long as she’s a good mum to him, spends time with him, shows him lots of love, feeds him well and he gets enough sleep, appearances don’t matter unless it upsets him in some way, e.g. his shoes are too tight.
It’s not your job to buy his clothes. So, if you want to treat him to some new things, better to ask her if that’s OK and what she’d prefer. My mum used to buy my kids clothes that weren’t at all to my taste (or theirs) and I felt terrible as I never dressed them in them.
If you’re worried she has depression or is struggling to cope financially, then love and support rather than judgement is the right response. Ask if she’s OK or if there’s anything you can do to help. Say you’d like to support her as you love her and know what it’s like, as you’re a mum yourself. Offer to treat your GC to some new clothes, but via a shopping trip together or looking for things together online. So, she/he can choose and you pay.
Ask if she’d like you to do any cleaning or housework while she’s away. But if she says no, then respect that.