(Related to MO Problem 9924.)
Suppose that $f$ is a polynomial of degree $d>0$ over a field of zero characteristic.
It is not difficult to see that if $Q$ is yet another, non-constant polynomial, such that the formal derivatives $(Qf)^{(k)}$ are divisible by $f$ for all $1\le k\le\deg Q$, then $f(x)=Cx^d$, up to a linear change of the variable. Notice also that $(Qf)^{(k)}$ cannot be divisible by $f$ for $k=\deg Q+1$.
Since there are no polynomials $f$ other than $f(x)=Cx^d$ such that $(Qf)^{(k)}$ are divisible by $f$, we relax the divisibility requirement, as follows. Suppose we are given a polynomial $f$ of degree $d>0$ and an integer $2\le m\le d/2$, and we want to find a non-constant polynomial $Q$ such that
$$ (Qf)^{(k)} = P_kf+R_k,\quad 1\le k\le\deg Q \tag{$\ast$} $$
with $\deg R_k<m$; that is, the derivatives $(Qf)^{(k)}$ are divisible by $f$ up to terms of degree smaller than $m$, for all $1\le k\le\deg Q$.
As an example, consider the polynomial $f(x)=x^d+r(x)$, where $e:=\deg r<m$. Letting $Q(x):=x^q$ with $q\le m-e$, for each $k\in[1,q]$ we have $$ (Qf)^{(k)} = C x^{q-k} f(x) - C x^{q-k}r(x) + (x^qr(x))^{(k)} = C x^{q-k} f(x) + R_k, $$ where $\deg R_k\le q+e-1<m$.
Are there any other pairs $(f,m)$, where $f$ is a polynomial of degree $d>0$ and $2\le m\le d/2$ is an integer, such that there exists a non-constant polynomial $Q$ satisfying ($\ast$)?
In fact, I would be interested even in a further relaxed version where $(Qf)^{(k)}=P_kf+R_k$ is required to hold, say, for $1\le k\le\frac12\,\deg Q$, or even for $1\le k\le 100$ only.
