The child of your first cousin is your first cousin, once removed.
Your child and the child of your first cousin are second cousins to each other.
First cousins once removed. Your child and your nephew are first cousins and there is one generation difference.
Your niece and your child are first cousins. There is a difference on one generation between the two. That would make them first cousins once removed.
The child of your father's first cousin would be your first cousin, once removed. Then when that person has a child, their child will be your first cousin, three times removed. A second cousin is the children of first cousins. For example, your child and your cousin's child are second cousins. Then their children will be second cousins, once removed to each other.
they would be cousins.
Your first cousin's child's child (your first cousin's grandchild) is yopur first cousin twice removed.
They would be second cousins. Your child and your grandniece's parent are first cousins. There are no 'removed' involved because the generation difference between the common ancestors (your parents) is the same.
The granddaughter is the niece's first cousin once removed.
Your child and your nephew are first cousins. Your child and your great nephew are first cousins, once 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.
Great grandniece or great grandnephew
second cousins to his cousin's children, you'd be third cousins
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.