Revs dropping frequently with fan on and even worse with A/C

[Mattyking]

I've just bought a 04 plate focus LX 1.6 and I've had an issue with the revs dropping to a point where it can stall when I come to a stop. This happens then the A/C is turned on but less frequently may happen with just the fan on alone. There seems to be a pulley 'thing' (I'm kind of new to cars, this is my first car!) going on and off constantly. It looks like some sort of pump (coolant?) or maybe part of the compressor. I really have no idea! There's a small crack in the coolant tank so there's a small amount of pressure being lost there, not sure if that will have anything to do with it. The crack is above the coolant level so there isn't any water being lost. I've ordered a new tank to replace it. Please see this short video on YouTube of what I've noticed, it's easier than me to try and explain 😂
https://youtu.be/vOavv-CZ0hI