Browning in this paper proves that if $f \in \mathbb{Z}[x]$ is an irreducible polynomial of degree $d \geq 3$ and $k$ is an integer $\geq 3d/4 + 1/4$, then we have $$\#\{n \in \mathbb{Z} \cap [1, X] : f(n) \text{ is $k$-free}\}\sim C_{1}(k, f)X$$ as $X \rightarrow \infty$ where $$C_{1}(k, f) = \prod_{p}\left(1 - \frac{\varrho(p^{k})}{p^{k}}\right)$$ and $\varrho_{f}(n) = \#\{a \pmod{n}: f(a) \equiv 0 \pmod{n}\}$. Is there anywhere that gives a similar asymptotic for the case when $f$ is reducible?
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