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 ${\left(x_{i+j}\right)}_{0\le i\le p,0\le j\le q}$ over $F$ have rank $\le r$ ? This question is classical, and the answer ($q^{2r}$ when $r\le \min \{p,q\}$) 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 $x_0,x_1,\dots ,x_{k-1}$ for some $k\le r\le \min \{p,q\}$, then the number of ways to choose the remaining $p+q-k+1$ entries $x_k,x_{k+1},\dots ,x_{p+q}$ such that the resulting Hankel matrix $\left( x_{i+j}\right) _{0\leq i\leq p,\ 0\leq j\leq q}$hasrank$\leq r$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.