Spaces of polynomials related to multiplier maps

Let $f(x) \in \mathbb{C}[x]$ of degree $n$. We attach to $f$ a $\mathbb{C}$-vector space $W(f)$ which consists of complex polynomials $p(x)$ of degree at most $n - 2$ such that $f(x)$ divides $f"(x)p(x) - f'(x) p'(x)$. The space $W(f)$ originally appears in Yuri Zarhin's solution towards a problem of dynamics in one complex variable posed by Yu. S. Ilyashenko. In this paper, we show that $W(f)$ is nonvanishing if and only if $q(x)^2$ divides $f(x)$ for some quadratic polynomial $q(x)$. Then we prove $W(f)$ has dimension $(n-1) - (n_1 + n_2 + 2N_3)$ under certain conditions, where $n_i$ is the number of distinct roots of $f$ with multiplicity $i$ and $N_3$ is the number of distinct roots of $f$ with multiplicity at least three.