How to get IDs of points attached to the cell?

Many thanks for the help!
I completed the above example according to your advice

paraViewData=FindSource('my_project.pvd')
SetActiveSource(paraViewData)
UpdatePipeline()
pythonData=servermanager.Fetch(paraViewData)
numberOfCells=pythonData.GetNumberOfCells()
from vtk import *
cellIds = vtkIdList() # cell ids store to
for cellIndex in range(numberOfCells): # for every cell
   pythonData.GetCellPoints(cellIndex, cellIds) # get ids of points of the given cell
   for i in range(0, cellIds.GetNumberOfIds()): # for every points of the given cell
      coord=pythonData.GetPoint(cellIds.GetId(i)) # get coordinates of the given point of the given cell, type: class 'tuple'
      x=coord[0] # get coordinates of the given point of the given cell, type: class 'float'
      y=coord[1]
      z=coord[2]

Regarding the definition of geometric centers of cells the Cell Centers filter reliability can be improved by adding the Clean to Grid filter before Cell Centers as explained here ParaView Cell Centers possible unstable program behavior

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