Adjust your expectations.
you are not going to have 5* luxury holidays to Mauritius on this budget but you might be able to camp or do a short break in uk
cheaper (supermarket brand ) rather than branded food.
meal plan for the week so you buy only the food you need.
adjust - rather than cinema and dinner out which is too expensive - you make it or buy a frozen pizza and popcorn and do a movie night at home
clothes - charity shop / vinted / supermarket - buying what you need not gimicky branded T-shirts etc. accept hand me downs from friends etc
birthdays - £100 buys a lovely big box of Lego or mechano , a load of play dough or loads of craft stuff.. check out deals in Argos and supermarkets - you can get a lovely present for far last than £100
library - new books every week for free.
never buy food out - invest in a good reusable coffee cup for you / make and take rather than buy . Buy good quality lunch boxes / water bottle for kids - if you go anywhere you take a picnic and drinks for them . Saves a fortune.
when mine were little we didn’t have £120 a week spare for ‘fun’ - we didn’t have anywhere near as much - think more £20 for all the things you mentioned… but mine always had everything they NEEDED (hugely different between need and want) and we had fun as we learnt to get by on what we had.
I did envy at times the people who went to soft play and for coffees and to big days out and had a car - but it just wasn’t within our budget and so I focused on what I did have - we are money poor - but I was able to research and find free / discounted things to do and set up activities at home that had minimal cost. We made going on the bus an adventure .
you can either get into big debt or just make the best of what you have and learn that ‘less is more’ and all that kids really need love and time not expensive clothes, holidays, stuff…
far better one present that they love and treasure and play with than a pile of stuff that sits and clutters up your house. Being less materialistic is a great thing to teach them as is teaching them to look after their things.
I never wanted mine to worry about money but I was honest with them about the realities of the world and that they couldn’t have everything they wanted instantly…
kids need to know the reality of life - make budget shopping fun.. show them the value of money and that you can only spend it once - ‘ we can spend £50 on pizza out’ - or we can buy pizza from the supermarket and still have money left over to have treats during the week and buy these multipack of crisps at and go swimming… you chose… mine now they are older are great bargain spotters and even at 5 knew the multipack of crisps from Tesco was better value than buying individually at the local shop…
it’s all about changing your mindset and making the best of what you have.
put it in perspective - some people are literally trying to feed and clothe and house their kids on your £120 ‘fun money’ per week - life really isn’t so bad when you look at it like that…