I will use this fact to prove that the group of automorphisms of a compact Riemann surface $C$ of genus $g>1$ is always faithfully represented in its action on the first homology $H_1(C)$.
Theorem: Every nontrivial automorphism $\varphi$ acts nontrivially on $H_1(C)$.
Proof: Assume otherwise, so that $\varphi$ is a nontrivial automorphism acting trivially on $H_1(C)$. Since $\varphi$ is holomorphic, it is orientation-preserving and thus acts trivially on $H_2(C)$ as well, and of course every self-map of $C$ acts trivially on $H_0(C)$. Thus the Lefschetz number of $\varphi$ is
$$L(\varphi)=\text{tr}(\varphi|H_0(C))-\text{tr}(\varphi|H_1(C))+\text{tr}(\varphi|H_2(C))=1-2g+1=2-2g.$$
The local description you give around a fixed point implies that every fixed point of $\varphi$ is isolated and has index 1. Thus by the Lefschetz fixed point formula, the Lefschetz number is equal to
$$L(\varphi)=\sum_{\text{fixed points}}1=\text{the }\#\text{ of fixed points of }\varphi.$$
But if $g>1$ the Lefschetz number $L(\varphi)=2-2g$ is negative, which is a contradiction.
I will now give a slightly different and perhaps more advanced application of this fact. I believe this argument goes back to Chevalley-Weil. Let $D\to C$ be an unramified Galois cover (aka a normal finite-sheeted covering space), and let $G=\text{Gal}(D/C)$ be the deck group, or Galois group. The finite group $G$ acts freely on the Riemann surface $D$ (with quotient $C$). We thus get an action of the group $G$ on the first homology $H_1(D;\mathbb{Q})$, and my goal is to describe this homology group as a representation of $G$. As before, let $g$ be the genus of $C$.
Theorem (Chevalley-Weil): There is an isomorphism $H_1(D;\mathbb{Q})\approx \mathbb{Q}^{\oplus 2}\oplus \mathbb{Q}G^{\oplus 2g-2}$ as representations of $G$.
Proof: A representation $V$ of a finite group over a field of characteristic 0 is determined by its character, the function $\chi_V(g)=\text{tr}(g|V)$. Thus it suffices to show that the character $\chi_{H_1(D)}$ coincides with the character $2\chi_{\mathbb{Q}}+(2g-2)\chi_{\mathbb{Q}G}$. Note that in the regular representation $\mathbb{Q}G$ every nontrivial $g\in G$ permutes the basis elements freely, so we have $\chi_{\mathbb{Q}G}(g)=0$ for $g\neq 1$ (and $\chi_{\mathbb{Q}G}(1)=|G|$). The character $\chi_{\mathbb{Q}}$ of the trivial representation is just the constant function 1.
First, the character $\chi_V(1)$ of the identity is just the dimension $\dim V$. If $h$ is the genus of $D$, multiplicativity of Euler characteristic gives $2-2h=|G|(2-2g)$, or $h=|G|(g-1)+1$. In particular, we have $\chi_{H_1(D)}(1)=2h=|G|(2g-2)+2$.
Now take $g\neq 1\in G$, and consider $g$ as an automorphism of $D$. Since $g$ acts trivially on $H_0(D)$ and $H_2(D)$, we have $$L(g)=1-\text{tr}(g|H_1(D))+1=2-\chi_{H_1(D)}(g).$$ But $G$ acts freely on $D$, so $g$ has no fixed points and the Lefschetz fixed point theorem gives $L(g)=0$. We conclude that $\chi_{H_1(D)}(g)=2$ for every nontrivial element $g\in G$, as desired. This shows that $\chi_{H_1(D)}=2\chi_{\mathbb{Q}}+(2g-2)\chi_{\mathbb{Q}G}$, which proves the theorem.
It is not hard to extend this argument to apply to ramified covers, since you can still describe the Lefschetz number of an automorphism by counting its ramification points (and this is what Chevalley-Weil actually did).
Best Answer
The degree of a $1$-differential $\omega$ on a Riemann surface of genus $g\geq2$ is at least $2.2-2=2$. Thus , any 1-differential has a zero.
Then The $q$-differential $\omega^q$ (whose local expression where $\omega=f(z)dz$ is given by $f(z)^qdz^q$ and for $q\geq1$ there is no ambiguity in this) has degree $(2g-2)^q\geq2^q>1$. Thus , every $m_q$ differential has zeroes (since all the $q$-differential belong to the $q$-canonical divisor class and their degrees are same).
This fact applied to the Wronskian in the question proves the existence of q-Weierstrass points on compact Riemann surfaces of genus $g$ for $g\geq2$ and $q\geq1$.(on the torus no Weierstrass points exist).