In your position I would stop worrying about benefits, and start thinking about how you can get yourself to a position where you don’t need to be thinking about benefits.
The “benefits trap” which you’ve discovered here is partly why it’s so difficult to get off benefits once you’re on them. But you’re not currently on them, and have decent savings, so you’re in a really good place for decision making.
How long have you had the £80k for? My guess is that your mum gave it to you to avoid/reduce inheritance tax.
You need a buffer, and some protection against possible inheritance tax, so you should keep £20k in the account. What can you do with the £60k to improve your financial situation?
Looking from the outside, the two obvious things are either invest in training so that you can earn more. Or pay off some of your mortgage and then re-mortgage so that you’re paying less each month.
Is your ex paying maintenance? If not, go to cms and make sure that you’re getting what you’re entitled to.
I know that your mum wanted the money for the kids, but she didn’t know that this would all happen. If she had, and she saw how tricky things were, she would definitely want you to spend it on improving things.