Related by the Hardy-Littlewood prime \(k\)-tuple conjectures, the sum \[ T_k(h) := \sum_{\substack{h_1, \dots, h_k \le h \\ \text{distinct}}} \mathfrak S(h_1, \dots, h_k), \] with \(\mathfrak S(\mathcal H)\) denoting the singular series \[ \mathfrak S(\mathcal{H}) = \prod_{p \text{ prime}} \frac{1-\nu_{\mathcal{H}}(p)/p}{(1-1/p)^k}, \] and \(\nu_{\mathcal{H}}(p)\) denoting the number of distinct residue classes modulo \(p\) occupied by the elements of \(\mathcal{H}\), has been studied for fixed values of \(k\). In the paper under review, the author studies sums of singular series for sets of size \(k\) with elements in \([1,h]\), where \(k\to \infty\) as \(h \to \infty\). More precisely, he shows that with \(k = O((\log h)^{1-\delta})\) for some fixed \(\delta > \frac{1}{2}\), and letting \(h, k \in \mathbb{N}\), then there exists a \(\beta > 0\), dependent only on \(\delta\), such that \[ T_k(h) = h^k + O(h^{k-\beta}). \] Furthermore, for arbitrarily large \(k\), the author shows that for \(k, h \in \mathbb{N}\) with no conditions on their relative growth rates, one has \[ T_k(h) \ll h^k \prod_{p \le k^3} \frac 1{(1-1/p)^k} \ll h^k (3\log k)^k. \] As an application, the author discusses the number of primes in the intervals of the form \((x,x + \lambda \log x]\), for which it is known that for any fixed \(\lambda>0\), assuming the Hardy-Littlewood prime \(k\)-tuple conjectures, \[ \lim_{x \to \infty} \frac{1}{x} \#\{n \le x : \pi(n+\lambda \log x) - \pi(n) = k\} = \frac{\lambda^k e^{-\lambda}}{k!}, \] where \(\pi(x)\) is the prime counting function. The author considers the case \(k \to \infty\), by studying the following moments \[ \frac 1x \sum_{n \le x} \left(\pi(n + \lambda \log x) - \pi(n) \right)^r. \] Under certain conditions, first he provides a relation between the above moments and the Stirling numbers of the second kind, and then obtains upper bounds for it.