Chen Jing Run proved that every large enough even integer is either the sum of two primes or the sum of a prime and a semi-prime (that is, the product of two primes). Golbach's conjecture states that every even integer greater than 3 is the sum of two primes. Harald Helfgott (happy birthday to him) proved that every odd integer greater than 5 is the sum of three primes.

What is the best known upper bound for the number $\overline{\mathcal{G}}(x) $ defined as the number of even integers below $x$ that are the sum of a prime and a semi-prime but not the sum of two primes?

Many thanks in advance.