This is exactly what a budget is for.
TBH I did originally go
at 10.5k but if that's seven full months of spending and includes everything aside from bills, it's 1.5k per month.
I bet £120 for the food shop doesn't cover top up shops, toiletries or cleaning products bought somewhere else like B&M//Savers, and random popping into the corner shop etc. So this will be more than the £500ish - I'd probably guess closer to £600/700 monthly. Because our food shop "normally costs" around €120, but our average monthly spend on food is actually €730. Which is more than 4.33x €120. You also probably forget to account for things like when it's school holidays and the children are at home, food costs increase.
So food shopping could easily be half of it or nearly. Then there are various essentials like kids' clothes, shoes, coats, petrol, stationery supplies for homework, printer etc, any home items that need replacing because they have broken or run out. If any of the kids have had a birthday, birthday presents, maybe party costs.
Then you mentioned fun spending as well like days out, takeaways, "general stuff".
I would go on the moneysavingexpert site and have a look at their free budget tool. It's easy to use and you can look at spending in different categories. I use YNAB now but it's not free. It might actually not be a bad idea to let DH have a go with something like this, or do it jointly - try it out over say a three month period and see how it goes, then discuss.
I think it can be easy to fritter money without thinking and if you're spending £500 without really knowing on exactly what, then it might well be possible to cut that down by 20% in order to give you back (as a family) the £100 monthly you were previously saving, which seems like a reasonable goal.
By looking at it like this then you can balance OK, how much fun money spending is reasonable vs preparing for the future. I don't think it's helpful or realistic to eschew all possible luxuries in the short term in pursuit of long term goals most of the time. But it does make sense to try and work out a good balance.