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.



Upper bound for the number of even numbers sum of a prime and a semi-prime not fulfilling Goldbach's conjecture
If I understand correctly, $\overline{\mathcal{G}}(x)$ is the usual exceptional set $E(x)$ except for possibly finitely many exceptions in the range $(4\cdot 10^{18},e^{e^{36}})$, where the Goldbach conjecture has not been verified, and an effective version of Chen's theorem does not apply. With that in mind, a classic result of Montgomery-Vaughan gives


$$E(x)<x^{1-\delta}$$


The best $\delta$ I know of is $0.121$, due to Wen Chao Lu in 2010.


