You do not want to cut out the circle, you want it to be meshed as part of the boundary. Probably you need to use the fragment command. See for instance Mesh, Submesh , Torus, Boundary - #5 by dokken
or
https://bthierry.pages.math.cnrs.fr/tutorial/gmsh/occ/basics/#boolean-operations-cad-like-a-boss-_
or the Gmsh documentation.