Let $\mathbb{F}$ be a finite field of odd characteristic and let $f:\mathbb{F}_{\leq d}[x,y]\rightarrow\mathbb{Z}$ map bivariate polynomials over $\mathbb{F}$ of degree at most $d\ll|\mathbb{F}|$ to the integers. Suppose $f$ is not identically zero. Suppose furthermore that for any line $\ell\subset\mathbb{F}^2$ and univariate $P$ defined on $\ell$: $$\sum_{Q:Q|_\ell=P}f(Q)=0,$$ where the sum is over bivariate $Q\in\mathbb{F}_{\leq d}[x,y]$ which, when restricted to $\ell$, equal $P$. Can one prove in this case, that $\text{Supp}(f)=\{Q\in\mathbb{F}_{\leq d}[x,y]:f(Q)\neq0\}$ must be large?

It is possible to show that $|\text{Supp}(f)|$ must be at least roughly $|\mathbb{F}|^2/d$ using that distinct $Q,Q'$ cannot agree on more than $d$ lines. However, I'm hoping a better bound (i.e., a bound which grows with $d$) might be possible as the linear conditions seem to give quite a bit more information than this "low-agreement" argument uses. A good bound in the special case when $f:\mathbb{F}_{\leq d}[x,y]\rightarrow\{0,\pm 1\}$ would also be interesting.