If there were such a thing as a foster cousin, your first cousin's foster cousin would not be related to you. However, if your family is a close one, you may find that person being treated as a member of the family, a cousin, and that is OK.
The child of your first cousin is your first cousin once removed.
They are cousins.
Your daughter and your first cousin are first cousins, once removed.
They are first cousins.
The children of your mother's first cousins are your second cousins. The children of your mother's second cousins are your third cousins. And so forth.
jew face
Your son and your niece are first cousins.
First cousin once removed.
Cousin once removed.
If your father and someone else's father are first cousins, then you and that someone else are second cousins.
second-cousin
You are one generation out of synch. That would make you first cousins once removed.
Your daughter and your first cousin are first cousins, once removed.