To not want my cousin's child at party

I would not offer to pay for a babysitter. Why? All the other parents of uninvited children would think that was unfair, unless of course no one else knew you were paying......before you know it you could be responsible for sorting out a crèche!
40 years ago, my sister took umbrage that her children were not invited to a family wedding, so refused invitation. Fast forward to her own daughter’s wedding, none of our family were invited, let alone children as it was a small do. Fair enough, but amazing how views can alter over the years.

No children means simply that

This is pandering to the child’s parents. Children are not invited and the 4 year olds family should respect this. Ne exception for this spoilt brat who’s family must accept that no means no. What is so difficult to understand about that ?
Pippa she is a single mum
I would imagine life can be hard enough coping alone so being critical is not what I would expect in a loving family.
And if I knew money was tight, what would be wrong with offering?
It’s called helping.

Hang on, or have I misunderstood this?

The OP , a niece, is organising a party in her home for her eighty year old aunt, and the guests are restricted to the siblings of the aunt, her friends and the other nieces and nephews, who are the OP's cousins.
She does not want to invite great nieces and nephews because that would be about twenty extra bodies and no room.(with me so far?).

Surely the single mum, daughter of her cousin, is a great niece and the child a great-great niece, therefore weren't on the guest list anyway?

One of my cousins, whose daughter is a single mum, has asked if they can bring her 4 year old as otherwise one of them will have to stay with her as usual
So it's the cousin who has asked presumably as she usually does the childcare?
Still a No though otherwise what about all the others?

I would tell your cousin that you can't change the invitation rule just for her, as it would cause upset to others.
If your cousin or her daughter can't find someone else to look after the child, then they just can't come.
Why do people have to make it all about themselves, when it's your Aunts birthday after all.

If the cousin (niece)'s daughter is available to attend the party to which she has not presumably been invited, she is available to look after her own daughter. Therefore no problem.
I do like to get things clear.

When we got married we said no children because one of DH's cousins had particularly obstreperous children. Come the day guess who turned up with her children? "I knew you wouldn't mind mine coming! " Didn't have time to do anything about it so we were stuck with them.

Sorry eazybee It was my aunt who asked if her daughter, my cousin, could bring her child. I accidentally typed cousin in my op.

Anyhow, another aunt has offered one of her grown up granddaughters (with the granddaughter's consent) as a babysitter for a few hours so problem solved

Can I just say as someone who brought up three kids on her own after a divorce (hate the term single mum -it takes two!) I never expected to get special treatment because I was on my own. Thankfully I had loads of supportive friends. Pleased everything is sorted for you have a lovely party!