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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.
Your first cousin's child would be your second cousin
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.
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 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 and your child are first cousins, once removed, to each other. Your second cousins and your child are second cousins, once removed, to each other.
Your first cousin's child is your first cousin once removed. Your second cousin's child is your second 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.
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.
Your child and your first cousin's child are second cousins. Your child and your second cousin's child are third cousins. It gets more complicated if you are talking about the child of a first or second cousins, etc, once removed or twice removed, etc.
All of your first cousin's children are your first cousins, once removed. All of your second cousin's children are your second cousins, once removed. It does not matter if the child is the first , second or tenth child. The English language has only the one term.
If your first cousin has a child, you and that child are first cousins, once removed, to each other. If your second cousin has a child, you and that child are second cousins, once removed, to each other.