A function field variant of Pillai's problem

In this paper, we consider a variant of Pillai's problem over function fields $F$ in one variable over $\mathbb{C}$. For given simple linear recurrence sequences $G_n$ and $H_m$, defined over $F$ and satisfying some weak conditions, we will prove that the equation $G_n - H_m = f$ has only finitely many solutions $(n,m) \in \mathbb{N}^2$ for any non-zero $f \in F$, which can be effectively bounded. Furthermore, we prove that under suitable assumptions there are only finitely many effectively computable $f$ with more than one representation of the form $G_n - H_m$.