Mcclegend said:

I’ve seen a few cars with the air filter removed when it goes for the mot.

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Is that supposed to help it pass, or just neglect on the owners part?

They are just tight gits that do not want to purchase a new filter and refit the old one after, not a good plan as a part clogged filter will result in higher fuel consumption.

LeadFarmer said:

Is that supposed to help it pass, or just neglect on the owners part?

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They told me it helps clear the engine and exhaust out, if you cars going to fail on emissions take the air filter out so there is no restrictions and give it a good blast to the test centre.

A good condition air filter will not restrict an engine, they are made the right size for the engine they are on. Some aftermarket filters are too small and while new are fine but performance soon falls off when they pick up dirt.
Many moons ago I was a Grasstrack racer and sometimes in the dry it was hard to even see beyond one's own bonnet. In these conditions it was vital to shake out the air filter often to maintain performance.
We once had a fellow racer who's car would take off like s--t off the proverbial but after about 2/3 laps would start to slow. Investigation reviled his sock air filter was clogging rather quickly, then when he stopped most the dirt fell off and motor would run fine again.