He sponsored an orphan; can he give him zakaah?

If this orphan who is in their care has no wealth or anyone
to spend on his needs, it is permissible for them to give him some of
their zakaah, because he comes under the heading of the poor. Allah, may
He be exalted, says (interpretation of the meaning):

“As-Sadaqaat (here it means Zakaah) are only for the
Fuqaraa’ (poor)

[al-Tawbah 9:60]. 

The basic principle is that anyone on whom you are not
obliged to spend, it is permissible for you to give zakaah to him. You are
not obliged to spend on this orphan, so it is permissible for you to give
zakaah to him. 

In Saheeh al-Bukhaari (1466) and Saheeh Muslim
(1466) it is narrated that Zaynab, the wife of ‘Abd-Allah ibn Mas‘ood (may
Allah be pleased with her), asked the Prophet (blessings and peace of Allah
be upon him): Is it acceptable for me to give charity (zakaah) to my husband
and the orphans who are in my care? He said: “Yes, and you will have two
rewards, the reward of (upholding ties of) kinship and the reward of giving
charity.’” 

Ibn Qudaamah (may Allah have mercy on him) said (2/271): If
there is among his family someone on whom he is not obliged to spend, such
as an orphan who is a non-relative, then the apparent meaning of Ahmad’s
words is that it is not permissible to give his zakaah to him, because he
would be benefitting by giving it to him because by doing so he is saving
himself some expense. But the correct view, in sha Allah, is that it is
permissible to give it to him, because he is included in the categories of
people who are entitled to zakaah, and there is no text, scholarly consensus
or sound analogy (qiyaas) to suggest that this is not allowed. It is not
permissible to exclude him from the general meaning of the text without any
evidence. End quote. 

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) was
asked: 

Is it permissible for me to give the zakaah of my wealth to
orphans whose legal shar‘i guardian is my father who is married to their
mother? 

He replied: If it was stipulated that your father should
spend on these orphans who are in his care, and your father is doing that,
then it is not permissible for you to give them zakaah, because they have no
need of that as a result your father spending on them. But if they are
staying with your father without it being stipulated that he should spend on
them, and they did not have any wealth from their own father [i.e., they did
not have any inheritance] then you may give them zakaah because they are
entitled to it.

End quote from Fataawa Ibn ‘Uthaymeen (18/353). 

And Allah knows best.