All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad is His slave and Messenger.
The father is obliged to spend on his children who are young and who do not have money to spend on themselves, and the mother or other relatives are not obliged to spend on the children so long as their father is able to spend on them.
Therefore, it is the right of this woman to refuse to foster her son whose father refused to spend on him and it is her right as well to give him to his grandmother.
As regards seeing the son, it is not due on his father to take him to the house of his mother so that she would see him; rather, the father must neither prevent the mother from seeing her son nor prevent the child from seeing his mother. For more benefit, please refer to Fatwa 84575.
Allaah Knows best.