Can he give to one of his children and not the others if he is poor?

Some of the
scholars said that it is permissible to show preference to some children in
giving, if there is a shar’i justification for that, such as if one of them
is disabled or has a large family or is preoccupied with seeking knowledge,
or if one of them is an evildoer or innovator, he may withhold some money
from him. 

Ibn Qudaamah
(may Allaah have mercy on him) said: If he singles out one of them because
he is in need, or he is chronically ill, or he is blind, or he has a large
family, or because he is preoccupied with seeking knowledge, or other
reasons, or he refrains from giving to one of them because he is an evildoer
or an innovator, or because he will use what he takes to disobey Allaah, or
spend it on such things, then there is a report from Ahmad which indicates
that that is permissible, because he said with regard to singling out one of
them for a waqf: there is nothing wrong with that if it is for a need, but I
dislike it if it is by way of showing preference. A gift is like a waqf. The
apparent meaning of his words may be understood as meaning that it is not
allowed to show favour to or single out one of them in any case. End quote
from al-Mughni (5/388) 

It says in
Fataawa al-Lajnah al-Daa’imah (16/193): What is prescribed with
regard to giving gifts to one’s children is to give them gifts equally; it
is not permissible to favour one over another unless there is a shar’i
justification, such as if one of them is disabled or  has a large family or
is busy seeking knowledge, or he may refrain from giving to one of his
children because of his evil or innovation, or because he will disobey
Allaah with what he takes. End quote. 

See also
al-Fataawa al-Kubra by Shaykh al-Islam Ibn Taymiyah (5/435) 

And Allaah
knows best.