Second cousin.
That would be your second cousin.
Your son and your 1st cousin's child are 2nd cousins.
They would be second cousins. They share a common Great Grandparent.
Your first cousin's child's child (your first cousin's grandchild) is yopur first cousin twice removed.
They would be second cousins, once removed. The descent is from a common ancestor. Thus 1st cousins share a grandparent as a common ancestor. 2nd cousins share a great-grandparent as an ancestor. If the descent from the common ancestor is not an equal number of generations, the inequality is expressed as 'removed' from the oldest generation being compared.
Your 1st cousin's granddaughter would be your daughter's niece.
Your nephew is your brother's or your sister's son. Your first cousin is also your brother's and your sister's first cousin. Your nephew and your first cousin are therefore first cousins, once removed, to each other. If your first cousin has a child, that child and your nephew are second cousins to each other.
Your first cousin's child is your child's second cousin. Your second cousin's child is your child's third cousin. Your third cousin's child is your child's fourth cousin. and so forth
They are also your 1st cousin.
There is no relationship in the English language that is called a "half cousin." Everyone who is your cousin is related to you through one and only one parent, so, unlike a "half brother" there can be no such thing as a "half cousin." If you have a first cousin (child of your aunt or uncle), then your child and that first cousin are first cousins once removed. If you have a second cousin, they your child and that second cousin are second cousins once removed.
Your second cousin is any child of you mother's or father's first cousins. These are people who are not your grandparents, your parents, your siblings or your first cousins but who share one or more great grandparents with you.
you and your cousin's children are 2nd cousins. ^^^ This answer is incorrect. your cousin's child would be your first cousin, once removed. (removed meaning a different generation.) Your second cousins share the same great-grandparents, but not the same grandparents. For example, your mother's cousin's child would be your second cousin.
Your daughter and your niece's child are 1st cousins once removed.