On the rank of Hankel matrices over finite fields

Given three nonnegative integers $p,q,r$ and a finite field $F$, how many Hankel matrices $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ over $F$ have rank $\leq r$ ? This question is classical, and the answer ($q^{2r}$ when $r\leq\min\left\{ p,q\right\}$) has been obtained independently by various authors using different tools (Daykin, Elkies, Garcia Armas, Ghorpade and Ram). In this note, we study a refinement of this result: We show that if we fix the first $k$ of the entries $x_{0},x_{1},\ldots,x_{k-1}$ for some $k\leq r\leq\min\left\{ p,q\right\}$, then the number of ways to choose the remaining $p+q-k+1$ entries $x_{k},x_{k+1},\ldots,x_{p+q}$ such that the resulting Hankel matrix $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$ has rank $\leq r$ is $q^{2r-k}$. This is exactly the answer that one would expect if the first $k$ entries had no effect on the rank, but of course the situation is not this simple. The refined result generalizes (and provides an alternative proof of) a result by Anzis, Chen, Gao, Kim, Li and Patrias on evaluations of Jacobi-Trudi determinants over finite fields.