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We find an upper bound for the sum $\sum_{x<n\leq2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$,where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in theadmissible set $\mathcal{H}=\{h_{1},...,h_{k}\}$, $m\geq 1$ and $x$ issufficiently large, with the same weights $w_{n}$ used in the Maynard's paper"Dense clusters of primes in subsets". The estimate will be uniform overpositive integer $k$ with $m+1\leq k\leq (\log x)^{1/5}$ and on admissible set$\mathcal{H}$ with $0\leq h_{1}<...<h_{k}\leq x$. The upper bound will dependon an integral of a smooth function and on the singular series of$\mathcal{H}$, which naturally arises in this context.

Weighted Average Number of Prime $m$-tuples lying on an Admissible $k$-tuple of Linear Forms