Does the mother have to provide for her children in the father's absence or in the event of divorce, and can she claim it back from him or not?

The father is obligated, by consensus, to financially support his poor, minor, and non-adult children, regardless of whether he has divorced their mother or not. If the father is absent and possesses accessible wealth, a reasonable amount should be taken from it to suffice them, provided that no harm results from this. If harm does result, the matter should be referred to the Sharia court, so the judge can mandate the maintenance for them. If the father does not have accessible wealth, then the mother is obligated to provide the financial support. She then has the right to reclaim what she spent from the father if she intended to do so at the time of expenditure.