What's the use of air gap in flyback transformer core?
Without the air gap, the magnetic field in the core would reach maximum and the core would saturate. On the opposite half cycle of the supply, this saturation would have to be destroyed by the new and opposite magnetising current, this is a waste of energy and it will limit the speed (frequency) at which the transformer can operate. Being a flyback transformer in a CRT circuit, the frequency of supply is the line frequency, that's pretty high.
it increases the reluctance Constant air gaps can be created in the core of certain transformers to reduce the effects of saturation. This increases the reluctance of the magnetic circuit, and enables it to store more energy before core saturation. This effect is also used in the flyback transformer.