In his paper "On the size of Kakeya sets in finite fields" (where the proof of the finite field Kakeya conjecture has appeared), Dvir also introduces the notion of a $(\delta,\gamma)$-Kakeya set, which is, essentially, a set $K\subset\mathbb F_q^n$ with the following property: there are at least $\delta(q^n-1)/(q-1)$ directions in $\mathbb F_q^n$ such that in each of these directions there is a line contining at least $\gamma q$ points of $K$. He then proves the following result:
Theorem. If $K\subset\mathbb F_q^n$ is a $(\gamma,\delta)$-Kakeya set, then $$ |K| \ge \binom{n+d-1}{n-1}, $$ where $d=\lfloor q\cdot\min\{\delta,\gamma\}\rfloor-2$.
To what extent does this answer the question?