For example, these characters have different radicals:
根 跟 恨 狠
But the part of the characters minus the radical is the same.
Does that part (艮 in this case) have a name?
On wiki, this part is defined as "phonetic". I am not familiar with these jargons in English, but I thinks this one is acceptable.
In your case,"艮"(gèn or gěn) is really a character.
Edit: Add Chinese names for this part
We call these characters “形声字”(in modern Chinese, many characters can be classified into this type). 形 means "form", referring to the radical, 声 means "sound", so I think "phonetic" is acceptable to call that part.
In Chinese, we call the "radical" 形旁 and the "phonetic" 声旁. Here, 旁(pang2) means a part.
If you start saying "phonophore" too, then there will be upwards of two people using the term. :-)