Counting algebraic points of bounded degree on curves

Let $X$ be a smooth projective curve over a number field $k$. Let $f\text{\hspace{0.17em}⁣}:X\to {\mathbb{P}}^1$ be a non-constant morphism that realizes the gonality of $X$. In this article we study the growth rate of $\{P\in X\left(\bar{k}\right)\mid [k(x):k]=\nu ,k(x)=k(f(x)),h(x)\le T\}\mathrm{.}$