See Grushko decomposition theorem.

Are the non-free factors of Grushko decomposition of a finitely generated convex–cocompact (but not cocompact) subgroup of $\operatorname{PSL}(2,\mathbb{R})$ finite?

In the cocompact case it is not true, since the group is not a free group and cannot be split into a non-trivial free product.

For convex–cocompact but not cocompact I know of particular examples with affirmative answer. Is it always the case?