I see no problem with the nest tbh and generally it is best not to move it and leave it for a week or 10 days up to 14 days, then when kits are stronger you can take them place somewhere in a warm soft place and check if it's wet and too soiled you can change all hay to dry and put the top layer back, all would be best to do when she's not there as well. If you have some absorbing material like wood pellets underneath it will stay dry and clean for longer so you will have to change it later anyway but when they are a bit older. First two weeks are critical, so best if you know she nurses she will take care of them.The way she built her nest its shoved to the way back, could that cause them to not feed properly as well? I have litter pans I used when they free roamed trying to potty train, would that be easier for her to feed them?
Attached is what she uses now.
I added some cotton to the bottom since they got scratched so the hay didn't make it worse.
That cotton wool lining looks just fine to me too.
Watch the smallest one if you see he's still wrinkled tomorrow morning maybe you want to help him, sometimes some babies are slower weaker than others and they may lose their chance to get enough milk since it all happens so quickly. Hopefully she has enough milk, there's only 4 kits so should be enough milk to feed them, sometimes they can have 8 kits or even 14 but with 4 it shouldn't be a problem.
Here's a video how you can help the weakest baby to get some extra milk. Please wait a bit maybe it will be full tomorrow after feeding, so you won't need this. just for your information and also you can see how they keep moving from one nipple to another.
This would be most natural way to help the weakest baby, so he could get some extra milk off its mother, please only use if it really needed, otherwise let her take care of her kits, just provide with good food, safe place and give her some privacy