OP I think you’re slightly missing some points.
You have a responsibility to your mum of course (sorry to hear about your DF), but that it’s not part of your homelife responsibilities that your husband is providing for. (And as you say that’s only become recently)
Imagine if your DH was on at you, as the one bringing in the income, for going to your mums and shopping every day rather than being at home doing X, Y, Z so that the things you need to do at weekend aren’t all necessary. I say that not as a dig that he brings the income but based on you saying you clean and he comes and messes it up.
Also you say you still did household chores when your DH was a SAHP, but it’s very different when you’re looking after babies/toddlers, it’s all consuming and you don’t have all the free time of someone with older children in FTE, as you know.
I can understand it being frustrating if your DH is leaving things lying round all the time, but think about it, if this is how he lives during the week and just tidies when he’s ready then he comes home to someone who does things differently and is constantly on him to move things there and then, there’s bound to be a clash. The best thing you can do for both of you is take a step back and leave him to do things in his time, if it doesn’t get done by the end of the day then sure say something, but chances are he will do it. Alternatively sit calmly with him and ask him how you find a compromise given that you live so differently during the week.
You’re going through a difficult enough time trying to support your mum after losing your dad, no doubt feeling the emotional drain, why get so worked up over small things like this when they don’t matter in the grand scheme of things. Leave it to your husband, he knows what he needs to do