  Generalized cluster algebras from orbifolds were defined by Chekhov and
Shapiro to give a combinatorial description of their Teichm\"uller spaces. One
can also assign a gentle algebra to a triangulated orbifold, as in the work of
Labardini-Fragoso and Mou. In this work, we show that the Caldero-Chapoton map
and the snake graph expansion map agree for arcs in triangulated orbifolds and
arc modules, and similarly for closed curves and certain band modules. As a
consequence, we have a bijection between some indecomposable modules over a
gentle algebra and cluster variables in a generalized cluster algebra where
both algebras arise from the same triangulated orbifold.
