No. The child of your first cousin is your first cousin, once removed.
The child of your first cousin and your child are second cousins to each other.
The child of your first cousin is your first cousin, once removed.The child of your second cousin is your second cousin, once removed.And so forth.
Your first cousin's child would be your second cousin
Your first cousin's child is your first cousin once removed. Your second cousin's child is your second cousin, once removed.
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.
Of course, your first cousin's child is related to you; you have ancestors who are also ancestors of your cousin's child. Your first cousin's child is your first cousin, once removed. Your second cousin's child is your second cousin, once removed.
Your son and your first cousin's son are second cousins to each other. Your first cousin's son is also YOUR first cousin, once removed.
Your first cousin's child is your first cousin once removed. Your second cousin's child is your second cousin, once removed.
Your child and your first cousin's child are second cousins. Your child and your second cousin's child are third cousins to each other.
Your first cousin's child and your child are second cosins. Your first cousin's grandshild and your child are second cousins, once removed. Your second cousin's child and your child are third cousins. Your second cousin's grandchild and your child are third cousins, once removed.
Your first cousin's child's child (your first cousin's grandchild) is yopur first cousin twice removed.
The child of you father's first cousin is your second cousin.
Your mother's first cousin's child is your first cousin, once removed.Your mother's first cousin's child is your child's second cousin.