On the representation of $k$-free integers by binary forms

Let $F$ be a binary form with integer coefficients, non-zero discriminant and degree $d$ with $d$ at least $3$ and let $r$ denote the largest degree of an irreducible factor of $F$ over the rationals. Let $k$ be an integer with $k \geq 2$andsupposethatthereisnoprime$p$suchthat$p^k$divides$F(a,b)$forallpairsofintegers$(a,b)$\mathrm{.}Let$R_{F,k}(Z)$denotethenumberof$k$-freeintegersofabsolutevalueatmost$Z$whicharerepresentedby$F$. We prove that there is a positive number $C_{F,k}$ such that $R_{F,k}(Z)$ is asymptotic to $C_{F,k}{Z}^{\frac{2}{d}}$ provided that $k$ exceeds $\frac{7r}{18}$ or $(k,r)$ is $(2,6)$ or $(3,8)$.