Riemann Surface, existence of meromorphic function.

Suppose the genus of $S$ is $g$ then consider the divisor $m\cdot p - q$. By Riemann-Roch theorem, we have $$ l(m\cdot p - q) - l(K - m\cdot p +q) = m - g $$ where $K$ is a canonical divisor. Let $m$ be large enough such that $deg(K - m\cdot p + q) = 2g - 1 - m < 0$ i.e. $l(K - m\cdot p + q) = 0$, then the dimension of $L(m\cdot p - q)$ grows linearly. Thus, there exists a meoromorphic function $f \in L((m+1)\cdot p - q)$ but $f \notin L(m\cdot p - q)$. That is, there exists a function $f$ that has a pole of order $m+1$ at $p$ and a zero at $q$ as you want.