It is a known result of Vinogradov that every sufficiently large odd integer $n$ can be expressed as sum of three primes (this has been improved recently for all $n\ge 7$, cf. here).
However, I do not know whether there is any result regarding the "structure" of some decompositions of this type. To be precise:
Question. Let $n$ be a sufficiently large odd integer. Do there exist primes $p,q,r$ such that $p+q+r=n$ with $p+q \le n/2$?