Let $C$ be a smooth projective curve defined over complex numbers. Recall that the gonality of $C$, $gon(C)$, is defined to be the minimal possible degree of a dominant morphism $C\to\mathbb P^1$.

I am interested in curves $C$ such that there is only one linear system $g_d^1$ satisfying $d=gon(C)$. Example of such curves are hyperelliptic curves of genus $\geq 2$ or trigonal curves of genus $\geq 5$, or more generally, $p$-gonal curves of genus $\geq (p-1)^2+1$ for $p$ a prime number. See, for instance, https://amathew.wordpress.com/2013/05/22/hyperelliptic-and-trigonal-curves/ for a reference.

I am interested in the converse of the above property. To be precise,

>  Let $C$ be a smooth curve such that there is only one linear system $g_d^1$ satisfying $d=gon(C)$. What can we say aboout $d,g=genus(C)$? Do we have any restrictions for relations of $d$ and $g$ (e.g., some nontrivial inequalities not predicted by Brill-Noether-Petri theory?)

Any comments are welcome!