What is meant by custody is keeping the dependent person and taking care of his interests.
The person in custody is the one who cannot take care of his own affairs independently and keep away from that which may harm him, because he has no discernment, such as a child or an adult who is insane or disabled.
It says in al-Mawsoo’ah al-Fiqhiyyah (17/301):
Custody of a small child is a necessity according the consensus of the fuqaha’. The same ruling applies according to the majority – the Hanafis, Shaafa’is and Hanbalis, and one opinion of the Maalikis -- with regard to an adult who is insane or disabled. End quote.
Al-Hajaawi said in Zaad al-Mustaqni’ (p. 206) in the chapter on custody:
It is essential to take care of children, the disabled and the insane. End quote.
The mother is more entitled to custody of her small or insane child than the father.
Al-Mardaawi said in al-Insaaf (9/416):
The most entitled of people to custody of the child or disabled person is his mother, without any dispute. End quote.
Ibn Qudaamah said in al-Mughni (8/192):
The child is only to be given the choice [i.e., between his parents, when he has reached the age of seven years] if two conditions are met:
(i) That both parents should be qualified to have custody. If one of them is not qualified to have custody, then it is as if that one is absent and the other one should be awarded custody.
(ii) That the child should not have a mental disability. If he has a mental disability, then he should stay with his mother, and not be given the choice because the mentally disabled person is like a child even if he is an adult. Hence the mother is more entitled to have custody of her mentally disabled child after he reaches adulthood. If the child is given the choice and chooses his father, then loses his reason, he is to be returned to the mother and his choice is rendered invalid, because he was given the choice when he was independent; if he loses his independence, this mother is more entitled to custody, because she has more compassion for him and is more able to take care of him, as is also the case in childhood. End quote.
Based on that, you are more entitled than your husband to have custody of your children who are mentally ill, so long as you do not get married again. But if you do get married again, then the father is more entitled to custody of his children.
Ibn Qudaamah said in al-Mughni (8/194):
If the mother marries again, her right custody is waived. Ibn al-Mundhir said: All of the scholars from whom I learned were agreed on that, because the Messenger of Allaah (blessings and peace of Allaah be upon him) said to the woman: “You will have more right to him so long as you do not marry again.” And because if she gets married, the rights of her husband will distract her from taking care of the child. End quote.
If the mother’s right to custody is waived as the result of her marriage, it passes to the next person. With regard to specifying who is more entitled after the mother, there is a difference of opinion among the fuqaha’. The correct view is that the father is the one who has more right after the mother, and this is the view of Shaykh al-Islam Ibn Taymiyah (may Allaah have mercy on him), and was also regarded as more correct by Shaykh Ibn ‘Uthaymeen.
See: al-Sharh al-Mumti’, 13/535.
Before all that it is essential to pay attention to the aim of custody, which is taking care of the affairs of the child who is in custody, looking after him and caring for him. If the father is going to neglect his children and their staying with their mother is better for them, they should stay with the mother. The one who should judge concerning this is the shar’i qaadi. As there are no sharee’ah courts in your country, you have no choice but to work out an agreement with their father concerning that or to refer the matter to the Islamic centre in your city, and they will do what they can to solve this problem.
With regard to your disposing of the money that they get:
If you are in need, there is nothing wrong with you doing that. But if your own wealth is sufficient for you and you do not have any need of their money, then it is better for you to do without their money, because Allaah says with regard to the orphan’s guardian consuming the orphan’s wealth (interpretation of the meaning): “and whoever (amongst guardians) is rich, he should take no wages, but if he is poor, let him have for himself what is just and reasonable (according to his labour)” [al-Nisa’ 4:6].
And because if you are poor, spending on your maintenance is obligatory on them, from their wealth.
We ask Allaah to make good easy for you wherever it is.
And Allaah knows best.