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