I do have sympathy, but would say YABU. Don't lie, but you and DH need to have a good chat about your finances. This will probably sound a bit harsh, but please hear me out.
This week you had no money for food. Yet you planned to buy a takeaway on Friday, and spend £300 on things for the kids, a lot of which you don't need, or could get cheaper. What something goes wrong with the car, or you get another unexpected bill? Either you will have been paid so much by then that that won't be a problem (in which case splashing out, a little, on the in laws wouldn't be either), or you risk not having enough money for food (again).
You have to prioritise. How much do you NEED for food (excellent suggestions earlier for cheap meals, homemade soup is also brilliant)? And bills? And essentials for the DC (shoes, not toys, books and playpen). Getting to work. That can't be touched for anything else.
With what is left, you should put some aside as a 'rainy day fund' - for unexpected expenses. Only after that look at non-essentials.
Your shopping list:
shoes & socks - £50 Clarks shoes, supermarket socks (much cheaper shoes available)
2 school shirts - £10
school jumper - £20 (assuming has to be expensive one with school logo)
name tag for bag why? make one / write on the bag.
babygros - £10 (for 6, supermarket)
vests - £5 (6)
socks - £3 (5 pairs)
leggings - £10 (for 4)
playpen - I think not essential. But if there is a good reason - could you borrow a travel cot from a friend? Last resort - buy for £40.
a few toys & books - Go to the library. Use DS's old toys, cardboard boxes, plastic bowls etc. If you must - something 'new' for £2 in a charity shop / second hand sale.
That's £110, without a playpen, £150 with. Possible to do a lot cheaper, especially if there are any nearly new sales on. Kids don't care whether something is second hand.
I do understand that you want to treat your DC, but when things are this tight that should be with money left at the end of the month (when you're about to be paid again), not money that you might need for something else.
If you don't already use it I'd recommend moneysavingexpert.com, there is lots of good advice on there.
I think the inlaws are a separate issue. So plan to do free, or very cheap things with them. Park, library, walks, playing in the snow, reading, boardgames. If they suggest anything else "sorry, we can't afford it". Maybe they are mean, and have no concept of skint; maybe they don't think you are, as when they visit you buy takeaway, DH ferries them round to expensive places etc.
Good luck for the weekend. At least they'll only be with you for 4 days!