Praise be to Allah.
If a man has issued the last of three talaaqs (divorces), or he has divorced her once or twice and her ‘iddah has ended, then she becomes a stranger (non-mahram) to him, and it is not permissible for her to be alone with him or to touch her or to look at her.
The relationship of an ex-husband with his ex-wife is like his relationship with any other non-mahram woman. The fact that they have children does not justify his looking at her, being alone with her or travelling with her. He can go out with his children without her being there, or she can be present with one of her mahrams, without doing any of the haraam things that we have mentioned.
Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him) said:
A thrice-divorced woman is a stranger (non-mahram) to the man like any other non-mahram woman, so the man does not have the right to be alone with her, just as he does not have the right to be alone with any non-mahram woman. And he cannot see of her what he cannot see of any non-mahram woman, and there are no special rulings concerning the relationship between them (other than the rulings governing all interactions between non-mahrams).
Al-Fataawa al-Kubra, 3/349.
With regard to accepting what the divorced husband spends on his children, there is no reason not to do so, even if he does not pray. She should remind her children of the importance of advising their father to pray, and may Allaah guide him through this advice.
If the mother fears that her children may be adversely affected by their kaafir father or that he may lead them to do something that Allaah has forbidden, it is not permissible for her to allow them to go out with him, because their going out with him is causing them harm.
And Allaah knows best.