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Let $\Lambda\left(n\right)$ be the Von Mangoldt function, let \[r_{G}\left(n\right)=\underset{{\scriptstylem_{1}+m_{2}=n}}{\sum_{m_{1},m_{2}\leqn}}\Lambda\left(m_{1}\right)\Lambda\left(m_{2}\right), \] \[r_{PT}\left(N,h\right)=\sum_{n=0}^{N}\Lambda\left(n\right)\Lambda\left(n+h\right),\,h\in\mathbb{N}\] be the counting function of the Goldbach numbers and the counting functionof the prime tuples, respectively. Let $N>2$ be an integer. We will find theexplicit formulae for the averages of $r_{G}\left(n\right)$ and$r_{PT}\left(N,h\right)$ in terms of elementary functions, the incomplete Betafunction $B_{z}\left(a,b\right)$, series over $\rho$ that, with or withoutsubscript, runs over the non-trivial zeros of the Riemann Zeta function and theDilogarithm function. We will also prove the explicit formulae in an asymptoticform and a truncated formula for the average of $r_{G}\left(n\right)$. Someobservation about these formulae and the average with Ces\`aro weight \[\frac{1}{\Gamma\left(k+1\right)}\sum_{n\leqN}r_{G}\left(n\right)\left(N-n\right)^{k},\,k>0 \] are included.

Explicit formula for the average of Goldbach and prime tuples representations