Does the time of the Adhan differ between governorates, and must each governorate adhere to its own timing, and why, given that most mosques call the Fajr Adhan according to Cairo time even if they are not in Cairo, and how exactly is the Adhan time determined, and is there any harm in delaying it slightly?

Prayer times are determined by cosmic signs. If cities are close to each other, they may unify their prayer times. However, if they are far apart, each province must have its own specific timing. The time for the Fajr adhan (definition) is at the appearance of the true dawn, and it is permissible to delay the adhan as long as the prayer time remains.