If a relative of mine is beyond a simple cousin and is actually a 2nd, 3rd (etc) cousin, I'd just say that they are an "umpteenth cousin" :-D It's a slangy way to indicate a high number (since "umpteen" doesn't actually exist as a number). People usually understand me when I do, and if not, I say that relative is "like a 2nd or 3rd cousin or whatever they are...some kind of cousin", lol
now the really complicated part is when you have to say "X is my 2nd cousin, once removed" or "twice removed", etc.; I still don't quite get how that works, but it involved just how far away from you (directly) that they're related; perhaps also pertaining to if related by blood or by marriage. Wait for more input on those points. ;-)