  We find an upper bound for the sum $\sum_{x<n\leq
2x}\textbf{1}_{\mathbb{P}}(n+h_{i_{1}})\cdots\textbf{1}_{\mathbb{P}}(n+h_{i_{m+1}})w_{n}$,
where $(h_{i_{1}},...,h_{i_{m+1}})$ is any $(m+1)$-tuple of elements in the
admissible set $\mathcal{H}=\{h_{1},...,h_{k}\}$, $m\geq 1$ and $x$ is
sufficiently large, with the same weights $w_{n}$ used in the Maynard's paper
"Dense clusters of primes in subsets". The estimate will be uniform over
positive 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 depend
on an integral of a smooth function and on the singular series of
$\mathcal{H}$, which naturally arises in this context.
