Praise be to Allah.
Zakaat al-fitr is obligatory upon anyone who possesses a saa‘ of food that is surplus to his needs and those of his dependents on the day and night of Eid.
It says in Daleel at-Taalib (p. 83): It is obligatory upon every Muslim who has food surplus to his needs and those of his dependents on the day and night of Eid.
He must give it on behalf of himself and on behalf of those Muslims whom he sponsors. If he cannot afford to give on behalf of all of them, then he must start with himself, then his wife, then his slave, then his mother, then his father, then his child, then the closest of those who would inherit from him. End quote.
The man must give zakaat al-fitr on behalf of his minor children who have no wealth. If they have wealth, then their zakaat al-fitr must be given from their own wealth. The same applies if they have reached adulthood; they must give their own zakaat al-fitr.
An-Nawawi (may Allah have mercy on him) said: If the child does not have any wealth, then his zakaat al-fitr must be given by his father, and that is obligatory upon the father, according to scholarly consensus, as was narrated by Ibn al-Mundhir and others. If the child has wealth, then his zakaat al-fitr must be paid from that wealth. This is the view of Abu Haneefah, Ahmad, Is-haaq and Abu Thawr. End quote from al-Majmoo‘ (6/108).
He also said (6/77): If the child is well off, then his maintenance and zakaat al-fitr should come from his own wealth, and is not due from his father or grandfather. This is the view of Abu Haneefah, Muhammad, Ahmad and Is-haaq. Ibn al-Mundhir narrated from some of the scholars that it should be paid by the father, and if he pays it from the child’s wealth, then he has sinned and is liable for it (i.e., he has to pay it back). End quote.
It has previously been explained that if the father has died, or he is poor and cannot spend on his children, and the mother is independent of means, then she is obliged to spend on the maintenance of those of her children who need maintenance.
In that case, if the mother is spending on her minor children, then their zakaat al-fitr is to be regarded as being like their maintenance and she must give it on their behalf, if they have no wealth.
See the answers to question no. 111811 and 149347.
It is permissible for a divorced woman to give her zakaat al-fitr and that of her children to her ex-husband, so long as he is poor, and the fact that he may take the money then spend it on his children, so it would go back to them, does not matter, for two reasons:
i. this is her wealth, and not that of the children;
ii. it does not matter if the wealth of the one who gives zakaah or charity comes back to him by some other means. Hence the more correct scholarly view is that the wife has the right to give zakaah to her husband, even if we assume that her husband will spend it on her and thus it will come back to her.
A similar case is if a person gives zakaah to a debtor who owes him money, then he takes it after it is paid back to him in settlement of that debt; this is permissible.
This is supported by the hadith of Umm ‘Atiyyah al-Ansaariyyah (may Allah be pleased with her), who said: The Prophet (blessings and peace of Allah be upon him) came to ‘Aa’ishah (may Allah be pleased with her) and said: “Do you have anything (to eat)?” She said: No, except that Nusaybah sent us some of the mutton which you had sent to her as charity. He said: “It has reached its destination [that is, it has reached us in a manner that is permissible].”
Narrated by al-Bukhaari (1494) and Muslim (1076).
This indicates that if charity is given to a needy person, it becomes his property and is like anything else that he owns: he may give it as a gift, sell it and so on, as he may dispose of anything else that he owns, with no differentiation.
Ibn Hajar (may Allah have mercy on him) said, commenting on this hadith: From this story, we may conclude that it is permissible for one who is owed money to take back from the poor person the same wealth that he gave to him by way of zakaah; and that a woman may give her zakaah to her husband even if he is going to spend on her from it. That is permissible so long as no condition to that effect is stipulated before giving the zakaah.
End quote from Fat-h al-Baari (5/242)
See also: Majallat al-Buhooth al-Islamiyyah (95/166).
And Allah knows best.