They are both idiomatic. (They are not really "idioms" but they are idiomatic).
I'm trying to lose weight. When I desperately want a snack, I eat an apple instead of potato chips or candy. Apples scratch my itch to eat something crunchy, as well as satisfying my sugar craving.
You are correct about the meaning of the second phrase. However, nobody ever admits to doing something like that. I can't imagine anybody saying "So-and-so is my pawn." It's usually used in the context of politics--world politics, national politics, or even just workplace politics.
It's a reference to the game of chess. In you aren't familiar with it, pawns are the shortest of the chess pieces; every other piece stands taller. Each player gets eight pawns. They have very little power. It is often judged that a queen is worth nine pawns. And players are very willing to sacrifice pawns, let the other player capture them in order to get some strategic advantage.
A recent example of use was a newspaper editorial with the headline, "X is Y's pawn," where X and Y were the names of world leaders. The meaning of course is that Y is playing a complicated game--figuratively, chess. And that while X may believe he or she is important, Y is manipulating X, treating X like a chess pawn--an unimportant piece, of little value, that can be easily sacrificed if the need arises.
A Google search finds me some other good examples. A 2008 book asked "First, was Yezhov just Stalin's pawn?" A 2013 book about US President Harry S. Truman. He was vice president under Franklin D. Roosevelt. The book described an editorial cartoon by saying "The implication was that he possessed no leadership ability of his own and would simply be Roosevelt's pawn."
Sometimes, people say "we're just pawns," meaning that even though, say, the CEO of a company praises us and says our jobs are important, in reality they treat us as chess pawns--unimportant pieces.