Harald's answer is perfect, but let me tell that using the circle method one can also prove similar results where $p_3$ runs through rather sparse sets (including sparse sets of primes). This is because exceptions to the Goldbach conjecture are relatively rare, i.e. $n+p_3$ will be a sum of two primes (less than $n$) for some odd $p_3<n$, even when $p_3$ is allowed to skip most values. See the work of Montgomery-Vaughan, Pintz, etc.

Harald's answer is perfect, but let me tell that using the circle method one can also prove similar results where $p_3$ runs through rather sparse sets (including sparse sets of primes). This is because exceptions to the Goldbach conjecture are relatively rare, i.e. $n+p_3$ will be a sum of two primes (less than $n$) for some odd $p_3<n$, even when $p_3$ is allowed to skip most values. See the works of van der Corput (1937), Tchudakoff (1938), Estermann (1938), Vaughan (1972), Montgomery-Vaughan (1975), etc.