We fix a lattice $\Gamma \subset \mathbb{C}$ and a period parallelogram of it and we pick nine distinct points $\omega_1, \dots, \omega_9 \in \mathbb{C}$ inside this parallelogram. We now insert the Weierstrass $\wp$ function and its derivative $\wp '$ into a homogeneous polynomial $f(x,y,z) = x^a y^b z^c$ of degree $d$ (i.e. $a+b+c = d$). So we get $f(\wp(z), \wp'(z),1) = \wp(z)^a \wp'(z)^b$ for a complex number $z \in \mathbb{C}$.
This is holomorphic away from the lattice points and has a pole of order at most $3d$ in the origin. Besides that, it is given that it has a zero of order at least $m$ at each of the points $\omega_1,\dots,\omega_9$.
The claim is now the following:
If $d < 3m$, then it follows from Liouville's Theorem (*) that $f(\wp(z), \wp'(z),1)$ is identically zero.
Whereas Liouville's Theorem (*) states that over any period parallelogram the double periodic meromorphic function $f(\wp(z), \wp'(z),1)$ has the same number of poles as zeros (counting multiplicities).
So if $p =$ number of poles and $n =$ number of zeros, we have $3d \geq p = n \geq 9m$ by Liouville. But on the other side we also have $3d < 9m$.
I don't know how I can conclude from this, that $f(\wp(z), \wp'(z),1) = 0$, so maybe someone else has an idea ?