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 jurists differed in opinion in regard to the ruling on spending on an adult child who is able to earn his living. The majority of them are of the view that it is not an obligation to spend on him while the Hanbali school of jurisprudence is of the view that it is an obligation to spend on him if he is poor. For more benefit, please refer to Fataawa 87752 and 92093.
Therefore, according to the view of the majority of the jurists you are not obliged to spend on him if he is able to earn his living. However, you should not give up spending on him so that you would gain the reward of giving charity and keeping ties with kinship. The Prophet said: “Giving charity to the poor is considered as charity, and giving charity to a relative is two things: charity and keeping ties with kinship.” [At-Tirmithi]
However, since he is at this age, he has the right to choose to stay with either you or his mother because he is no longer at the stage of fostering and he has become a man. For more benefit, please refer to Fataawa 89512 and 125277.
In case his mother really asks that he stays with her, then being kind to his mother has coincided with being kind to his father. The scholars stated that the mother comes in priority over the father in regard to their children being kind and dutiful to them when there is discrepancy between both rights; as the Prophet gave the mother three-quarters of the kindness of the children. Indeed, the Prophet was asked by a man: “Who deserves my good company more?”, and he replied: “Your mother.” The man asked: “Then who?” The Prophet said: “Then your mother.” The man asked again: “Then who?” The Prophet again replied: “Then your mother.” The man asked again: “Then who?” Thereupon, the Prophet said: “Then your father.” [Al-Bukhari and Muslim]
Nonetheless, we advise your son to convince his mother to let him be with you for some days and stay with her for some other days so that he would combine between being kind and dutiful to his father and to his mother.
Finally, we advise you both as his parents to help him be kind and dutiful to both of you as this will help him in his good nurturing.
As regards what the law may take as procedures towards this issue, then you should consult the specialists in this regard.
Allaah Knows best.