Assuming that the new parent and his cousin are first-cousins, the newborn daughter is the cousin's first-cousin once removed.
Your cousin's daughter is your first cousin once removed. Your daughter and your cousin's daughter would be second cousins.
the first cousin and daughter would be second cousins.
If by cousin you mean first cousin, your daughter and your (first) cousin's son's are second cousins to each other.
Your first cousin's daughter is first cousin once removed to your grandson, and he to her. Your second cousin's daughter is second cousin once removed to your grandson, and he to her.
Your first cousin's daughter is your first cousin, once removed.
Your Second Cousin's daughter is your Second Cousin, Once Removed.
Your daughter and your first cousin are first cousins, once removed, to each other. Your daughter and your second cousin are second cousins, once removed, to each other.
If your cousin is your 1st cousin then the daughter would be your 2nd cousin. Sorry, WRONG-OOO. YOUR COUSIN'S DAUGHTER IS YOUR FIRST COUSIN, ONCE REMOVED. Your children and your cousin's children would be second cousins.
Your first cousin's daughter and you are first cousins, once removed. Your first cousin's children and your children will be second cousins to each other.
They would be second cousins.
Your daughter and your first cousin's daughter are second cousins to each other. Your daughter and your first cousin's granddaughter are second cousins, once removed, to each other. Your granddaughter and your first cousin's granddaughter are third cousins to each other.
You and your first cousin's daughter are first cousin, once removed, to each other. Your child and your first cousin's child are second cousins to each other.