I am not sure if this is a valid question but posting any way:

Say I am over $\mathbb{F}_{p}$ for a prime $p$.

I have a curve of form $x^{2} = f(y)$ where $f(y)$ has an unknown form (and hence degree). How many points do I need to know on the curve to estimate the genus of the curve?

$A.)$We also have the additional constraint that $f(y)$ has atmost $2k$ non-zero coeffients where $k$ is a constant. Assume that a bound to the degree is known.

$B.)$Assume $2k << y$-degree of the equation.

$C.)$Assume the coeffients of the highest half of the terms are $+1$ and the lowest half of terms are $-1$. (Just an artificial example - but this tells that one possibly may be able to get the genus without getting the coefficients. For a concrete realization of the artificial example, look at error correction codes over $3$ alphabets $\{ \pm1, 0 \}$. The errors can be in only $2k$ coordinates and I also know the errors in the top half will be $+1$ and the lower half will have errors with $-1$).

How many points do you need? If degree bound is $D$, then would $O(\log^{h(k)}{D})$ points suffice where $h(k)$ is independent of $D$ and of the curve and is fixed for a fixed $k$?