Assuming both children are on the same side of the family, your child's first cousin is a second cousin to your first cousin's child. Your child is also a second cousin to your first cousin's child.
Your child can have first cousins who are not related to your first cousin and thus not related to your first cousin's child.
You and your uncle's child are first cousins to each other. You and your uncle's grandchild are first cousins, once removed, to each other.
Your child and your sister's child share a common grandparent and are first cousins. Your grandchild and your sister's grandchild share a common great grandparent and are second cousins. There is a generation out of synch, so they would be second cousins once removed.
Your child and your sister's children are first cousins. Your grandchild and your sister's children are first cousins once removed.
Your grandchild and your niece are first cousins, once removed.
The grandchildren would be the parents, aunts and uncles, and first cousins once removed, of the great grandchildren.
You would be Second Cousins
You are the grandchild of your grandfather. If that grandfather had children other than your parent, and that child had a child - that person is your first cousin and also a grandchild of your grandfather. Your first cousins are, by definition, all those who are not your siblings but are also grandchildren of one of your grandparents. Remember that you have two sets of grandparents. Those first cousins who are related to you through one pair of grandparents are not related to your other grandparents at all.
The grandchild of your father's first cousin and your child (your father's grandchild) are third cousins to each other. In the US, no jurisdiction prohibits such marriages.
Your daughter and your sister's child are first cousins. Your daughter and your sister's grandchild are first cousins, once removed.
Great nephew. Your children are the first cousins of your niece and your children would be first cousins once removed. Your grandchildren and your niece's child would be second cousins.
Your niece's child and your grandson's child (your great grandchild) are second cousins, once removed, to each other.
Your child and your nephew are first cousins, so your grandchild and your nephew are first cousins once removed.