The method of annihilating positrons in positron annihilation lifetime measuring mode is applied to study mechanism of rare-earth doping in Ga-codoped arsenic selenide As2Se3 glass modified with Sb. The atomic-deficient structure of parent As2Se3 glass is imagined as containing positron-trapping sites in the form of free-volume voids within cycle-type arrangement of corner-sharing trigonal AsSe3/2 pyramids, composed of atomic-accessible geometrical holes arrested by surrounding atomic-inaccessible Se-based bond-free solid angles. The Ga codoping in As2Se3 glass causes gradual decrease in trapping rate and fraction of trapped positrons due to agglomeration of free-volume voids. Partial As-to-Sb replacement in Ga-codoped As-Se glasses leads to better stability against crystallization processes and possibility to further rare-earth doping without principal changes in the type of positron-trapping defects. Effect of 500wppm of Pr3+ in Ga-2(As0.28Sb0.12Se0.60)(98) glass is explained in terms of competitive contribution of changed occupancy sites in Ga-modified glassy network available for rare-earth ions and annihilating positrons.