Improve mapping data on a Function by coordinates

[dokken]

Note that you can use mesh.geometry.input_global_indices together with dolinx.mesh.entities_to_geometry to get the original node index in your Gmsh file.
https://docs.fenicsproject.org/dolfinx/v0.8.0/python/generated/dolfinx.cpp.mesh.html#dolfinx.cpp.mesh.Geometry_float32.input_global_indices

What is the difference between compute_incident_entities and entities_to_geometry?

entities to geometry computes the geometry (node indices) of an entity of any dimension (vertex, Edge, facet, cell). The nodes are listed per entity. Compute incident entities takes in a list of entities, and returns a single list of all entities of dimension d1 connected to these. In the case of a linear geometry, all nodes = vertices, and the connectivities might be the same for your call (not always guaranteed).

This is assuming you can extract the index from Gmsh in step 1.