OP, as you've seen, the MN first wives club pounce on these threads. Ignore the ridiculous. You're the arse and the poor mother and any of her children can do no wrong. Your children and family should go without, as long as the adult your DH is the father of can maintain his handouts. Age 20. In case one of them hasn't piped up with it already, this is all your fault because you knew he had kids when you met him, and the other pearler that's always pedalled..."OP, were you the other woman???" Now get in the bin Cruella. 
Ok, now that's dealt with, you can look at the genuine responses.
There's a huge difference between the support of your parents once becoming an adult, and paying for a 20yr old adult who chooses not to work the same you would a 9yr old totally dependent child, to live day to day.
No, at twenty, if he chooses not to work and to do a college course, this is not down to DH to finance. Should my son choose at twenty, to still live at home, with a college course for 2/3 days a week, he's got the rest of the week to get a job to contribute to his life as an adult!!! Why should he be exempt at 20 for having to take any financial responsibility for himself?
If he needs occasional expenses, books, materials, then ad hoc, I'd have no problem paying directly for those things. But to just handover (estimate here as you say it's hundreds a month) £150 a week to a grown 20yr old who doesn't want to get himself a job? Please 
You can't blame the ex. She's just going for what she can get. Why give a flying fuck that your children might go without so DH can finance this work shy adult. And this is the way it should be according to the first wives club. The children don't matter. Unless it's a first wife child. That child, and solely that child matters. Yours are second class citizens, and it's ok for them to be disadvantaged and left with whatever crumbs remain. You must give her child every consideration whilst she happily watches yours suffer to the contrary. So of course if there's money for her to still take, don't expect anything less. God forbid she tells her adult son to take any accountability for himself.
If the son chooses as an adult not to financially support himself, it's not down to DH to pay for another adults living expenses.
This adult needs to take responsibility for himself.
My parents are going to buy (me) my forthcoming DC their pram. That's lovely of them, and normal parental support because I'm an adult. I think the last time they "supported" me was about 5 years ago, they treated me to a canteen of cutlery. What's not normal is demanding an entitlement to £150 a week, every week, because I don't fancy working and, well, they should pay for me. Because genetics.