A child of two first cousins is the child of the parents and they are the child's mother and father. As the child of one first cousin, the child is also the first cousin, once removed, of the other parent. That, however, is of no importance. The closer relationship of parent and child is the one that really matters.
Your cousins child is your first cousin once 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.
They are cousins.
Your child and your first cousin's child are second cousins. Your child and your second cousin's child are third cousins to each other.
Your son's child and your nephew are second cousins.
You are second cousins
they would be cousins.
Your niece's child and your grandson's child (your great grandchild) are second cousins, once removed, to each other.
The relationship between your sister's (or brother's) child and your cousin is the same as the relationship between your child and your cousin. Your child and your first cousin are first cousins, once removed. Your child and your second cousin are second cousins, once removed, and so forth.
Your child and your sister's daughter are first cousins. Your grand child and your sister's daughter are first cousins, once removed.
Second cousins.
Your son and your 1st cousin's child are 2nd cousins.