Let $X$ be an open Riemann surface (or more specifically a compact Riemann surface minus one point). Let $\omega$ be a closed holomorphic $1$-form on $X$.
For background, a closed Riemann surface $Y$ has holomorphic de Rham cohomology group $H^1_{dR}(Y)$ of dimension $g$. It is possible to find a basis $\{\sigma_i\}$ of $H^1_{dR}(Y)$ such that $\int_{A_j} \sigma_i=\delta_{i,j}$ where $A_j$ is one of the $g$ $A$ cycles on $Y$. For a closed $1$-form $\omega$ on an open Riemann surface, to show exactness, it suffices to show that the integral of $\omega$ along all $A$ and $B$ cycles and around all punctures is zero.
I am particularly interested in question B below.
A) What can be said about the cohomology class of $f\omega$ for $f$ a global holomorphic function on $X$?
B) Does there exist a global holomorphic function $h$ such that $h^2\omega$ is exact? (The particular problem I am interested in requires the square.)
I suspect the divisor of $f$ or other information about the zeros of $f$ maybe enough information to determine the cohomology class.
I would also be interested in knowing an answer to these questions in the algebraic category, that is if $X$ is an affine curve over $\mathbb{C}$ and considering Kahler differentials instead. However, I suspect the holomorphic category is easier to work with. For instance, any vector bundle on an open Riemann surface in the holomorphic category is trivial, which is not true in the algebraic category.