In my opinion, as we anneal the material stresses are removed in the material and bond becomes stiffer relative to stressed states of grains in material that's why relatively more energy is required to get phonon vibration of same material after annealing, results into blue shift.
The annealing ameliorate the ZnO crystal quality. So, it's normale that the position of E2(H) shifted after annealing. I advice you to compare your E2(h) with ZnO crystal standard.