What is the minimum size of a subset $S \subseteq \mathbb{F}_p^n$ such that for all directions $a \in \mathbb{F}_p^n$, there is a line in direction $a$ that intersects $S$ in at least $C$ points?

If $C = p$ this is exactly the finite field Kakeya problem, and we know $|S| \ge C_n p^n$ by the proof of Dvir in On the size of Kakeya sets in finite fields. However, I am curious what bounds we can show if e.g. $C$ is much smaller than $p$, eg. $C = \log^{O(1)}(pn)$.

I believe by random sampling, one can show an upper bound of something like $\lvert S\rvert \le p^{(1-1/C)n}$, which is trivial for $C = \log^{O(1)}(pn)$. A lower bound of $\lvert S\rvert \ge p^{(n-1)/2}$ is also clear, because the points in $S$ determine at most $\lvert S\rvert^2$ lines.

Also, in Dvir's paper, he also handles the case where $C = p^{1-\epsilon}$, but I am asking for even much smaller values of $C$. So is the truth closer to $p^{n/2}$ or $p^n$?