Is the pump fitted in the loft ? if it is then that is why there is air vents.
Make and model of pump would be helpful.
With out knowing this I can only guess.
Your air is coming from the hot water cylinder and poss the cold water tank especially if you have a small storage of water.
you should have an air free supply to the hot side of the pump eg Surrey Flange, Essex or the best one Techflange.
if the pump is in the loft plumb in an anti-gravity loop.
the air in the pump needs to be rectified but is not the problem.
The issue you have buy the sound of it is something called hunting where the water in the pipework is moving back and forth this is common with long pipe runs you will find that the hot side is the side that is effected normally.
test by letting the pump get into that state where it is on off then isolate the hot side of the pump if this stops it, then fit a non return valve on the outlet side of the pump.