Factoring polynomial values into smaller polynomial values not divisible by other values

I would like some help with this question: let $S$ be a sparse subset of $\mathbb {N }$. Let $M$ be a subset of $S$ such that if $m\in M$ and $sa=m$ with $s\in S$ implies that $s=m$ and $a=1$. Let $S(x)$ be the number of represnetations of elements of $S$ less than x. We say that $c(n)$ is the number of representations of n as $m_1m_2m_3\ldots m_n$ with all $m_k\in M$. What are asymtotics for $$A_{d}(x,a)=\sum_{\substack{n\le x\\n\in S\\n\equiv 0\mod d\\a|d}}c(\frac{n}{a})1_{S}(n).$$ Moreover we know that $\sum_{\substack{n\equiv 0 \mod d\\n\in S,n\le x}}1=g(d)|S(x)|+r_d$. We also know that if the product $\prod_{n\in S}\frac{1}{1-\frac{1}{n^{s}}}$ converges if $s> \sigma$ then $\sum_{\substack{n\le x}}c(n)=o(x^{\sigma+\varepsilon})$ as each coefficent of this series is greater than the number of factorizations into $M$ by theorem K of this paper. I would like a more acurate asymtotic or bound with a somewhat resonable error term. Hopefully in terms of $S(x)$. Thanks for the help.