Explicit formula for the average of Goldbach and prime tuples representations

Let $\mathrm{\Lambda }\left(n\right)$ be the Von Mangoldt function, let $$r_G\left(n\right)=\mathrm{\Lambda }\left(m_1\right)\mathrm{\Lambda }\left(m_2\right),$$ $$r_{PT}(N,h)=\sum _{n=0}^N\mathrm{\Lambda }\left(n\right)\mathrm{\Lambda }(n+h),h\in \mathbb{N}$$ be the counting function of the Goldbach numbers and the counting function of the prime tuples, respectively. Let $N>2$ be an integer. We will find the explicit formulae for the averages of $r_G\left(n\right)$ and $r_{PT}(N,h)$ in terms of elementary functions, the incomplete Beta function $B_z(a,b)$, series over $\rho$ that, with or without subscript, runs over the non-trivial zeros of the Riemann Zeta function and the Dilogarithm function. We will also prove the explicit formulae in an asymptotic form and a truncated formula for the average of $r_G\left(n\right)$. Some observation about these formulae and the average with Ces`aro weight $$\frac{1}{\mathrm{\Gamma }(k+1)}\sum _{n\le N}{r}_G\left(n\right){(N-n)}^k,k>0$$ are included.