We give a $q$-enumeration of circular Dyck paths, which is a superset of the classical Dyck paths enumerated by the Catalan numbers. These objects have recently been studied by Alexandersson and Panova. Furthermore, we show that this $q$-analogue exhibits the cyclic sieving phenomenon under a natural action of the cyclic group. The enumeration and cyclic sieving is generalized to Möbius paths. We also discuss properties of a generalization of cyclic sieving, which we call subset cyclic sieving. Finally, we also introduce the notion of Lyndon-like cyclic sieving that concerns special recursive properties of combinatorial objects exhibiting the cyclic sieving phenomenon.