The child of your first cousin, once removed, is your first cousin, twice removed, if your first cousin once removed is the child of your first cousin.
If your first cousin once removed in the first cousin of one of your parents, the child is your second cousin.
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If your first cousin once removed is your parent's first cousin, then your first cousin once removed is your child's first cousin twice removed.
If your first cousin once removed is the child of your first cousin, then your first cousin once removed is your child's second cousin.
Your father's brother's child is your cousin. A first cousin is anyone who shares one or more common grandparents with you.
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 father's niece is your first cousin. Her son is your first cousin, once removed.
Your mother's nephew is your First Cousin.
Your uncle's daughter is your first cousin. In the English language, it does not matter whether the uncle is a maternal uncle or a paternal uncle. All the children of your aunts and uncles are your first cousins.