Your father's (or mother's) first cousin's son (or daughter) is your second cousin. Second cousins are people who do not have the same parent or grandparent, but have the same great-grandparent.
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It would technically be considered your half- cousin since you do have the same grandmother but a different grandfather.
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.
The aunt of your first cousin's child is your first cousin's sister, who is also your first cousin.
Your mother's first cousin is YOUR first cousin, once removed. The child of your mother's first cousin is YOUR second cousin.
Your first cousin's child is your first cousin once removed.