It's the icing on the cake.
Here, is on the cake an adjunct or a complement of the noun icing?
A complement is something that is required to complete a grammatical structure.
An adjunct is something that adds something, but is not essential for the grammar.
In a sentence "It is a red cat", a red cat is a complement, because "It is" is not complete without it. But red is an adjunct, because "It is a cat" is complete.
So you can answer your own question if you can say whether "It's the icing" is grammatically complete or not.
In this case, "icing on the cake" is an idiom. It only has the meaning that it does if all of the words are included. As such, it functions as a single noun in a sentence.
You can't separate part of that noun and maintain any of its meaning. So asking if "on the cake" is an adjunct or a complement of "icing" is like asking if "ngbird" is an adjunct or a complement of "mocki". I.e. I don't think this is a meaningful question.