Positron annihilation spectroscopy (PAS) was used to examine the effect of defined Cd-rich and Te-rich annealing on point defects in Cl-doped CdTe and Ge-doped CdZnTe semi-insulating single crystals. The as-grown crystals contain open-volume defects connected with Cd vacancies . It was found that the Cd vacancies agglomerate into clusters coupled with Cl in CdTe:Cl, and in CdZnTe:Ge they are coupled with Ge donors. While annealing in Cd pressure reduces of the density, subsequent annealing in Te pressure restores . The CdTe:Cl contains negatively-charged shallow traps interpreted as Rydberg states of A-centres and representing the major positron trapping sites at low temperature. Positrons confined in the shallow traps exhibit lifetime, which is shorter than the CdTe bulk lifetime. Interpretation of the PAS data was successfully combined with electrical resistivity, Hall effect measurements and chemical analysis, and allowed us to determine the principal point defect densities.