On the level of distribution of Goldbach primes and its applications

We prove that, for almost all even integers $N>0$, the set of Goldbach primes $\mathbb{P}\cap (N-\mathbb{P})$ has a level of distribution $1\mathrm{/}6$. As applications, we show that almost all even integers $N>0$ can be written as the sum of two primes $p_1,p_2$ such that $p_1-p_2+1\in {\mathbb{P}}_4$. We also prove an analogous result with $2p_1{p}_2+1\in {\mathbb{P}}_{13}$ for almost all integers $N>0$ with $6\mid N$.