It doesn't exactly "cycle", and in fact it's too dangerous to board with strong allergies due to how long the air circulates in a plane[1]. Anything else is airline PR trying to prevent total shutdown. A good fitted mask gives considerable increase in safety factor, though, so if you need to fly, wear one.

[1] If you have really strong allergies for example for peanuts, you're obliged to inform the airline about it, and then the final decision on whether you fly or not belongs to plane captain. If they agree, it usually involves things like banning peanuts from the flight (no matter where bought), and keeping a steward(ess) with epipen close by for the whole of the flight. At least one person got banned from a low cost airline in Europe when they opened their airport-bought peanuts on such a flight (after multiple repeated announcements that it's disallowed). The air isn't cycled enough to prevent peanut dust from causing anaphylactic shock on the other side of an 737.