Air gap in switching power supply transformer
Air gap just adds more reluctance - increase of magnetic resistance. Like adding  a resistor in the circuit to limit the current (flux) with the scope to avoid core saturation. 

Adding the gap does not filter out the DC component, it behaves the same with AC, too. 

Take some examples how o calculate the HF core step by step, you will then come to the point when to use or not the air gap. There are no cons/pros, once you have to add the gap - it just can't work without it. And it has to be quite exact, too.