Let $C$ be a smooth algebraic curve. Suppose there is a branched covering map $\pi:C\to P^1$ where $P^1$ denotes projective line. Say $\pi$ is branched over $a_1,\dots, a_n$ with branching order $\geq 1$.(i.e. Locally, the chart is at least 2-fold covering.) Say $a_2$ has branching order $5$. Denote $\{f_k\}=\{\pi(a_i)\}$.
$\textbf{Q:}$ Is there an automorphism of $C$ s.t. I can make $a_2$ branching order 1? In other words, given $C$, can I twist the map $\pi$ by automorphisms of $C$ and $P^1$ s.t. branching order $a_2$ is changed from $5$ to $1$ but this may produce new branching points? This should look like rewriting the polynomials.