Partitions into polynomial with the number of parts in arithmetic progression

Let $k, a\in \mathbb{N}$ and let $f(x)\in\mathbb{Q}[x]$ be a polynomial with degree ${\rm deg}(f)\ge 1$ such that all elements of the sequence $\{f(n)\}_{n\in\mathbb{N}}$ lies on $ \mathbb{N}$ and $\gcd(\{f(n)\}_{n\in\mathbb{N}})=1$. Let $p_f(n)$ and $p_f(a,k;n)$ denotes the number of partitions of integer $n$ whose parts taken from the sequence $\{f(n)\}_{n\in\mathbb{N}}$ and the number of parts of those partitions are congruent to $a$ modulo $k$, respectively. We prove that there exist a constant $\delta_{f}\in\mathbb{R}_+$ depending only on $f$ such that $$p_f(a,k;n)=\frac{p_f(n)}{k}\left[1+O\left(n\exp\left(-\delta_{f}k^{-2}n^{\frac{1}{1+{\rm deg}(f)}}\right)\right)\right],$$ holds uniformly for all $a,k, n\in\mathbb{N}$ with $k^{2+2{\rm deg}(f)}\ll n$, as $n$ tends to infinity.