My guess would be it's this but I think there are 2 cats on the fsi's. A non OEM cat and the complicated fsi engine in my case didn't go that well together and threw up the eml light from time to time. Mine was the rearmost cat that was replaced. I put it down to not enough back pressure and/or the (I think) 2 exhaust sensors not reading correctly. Didn't actually seem to cause an issue just annoying having the light on.
Yes, the FSI has two cats. The first one is fitted directly after the exhaust manifold, where the downpipe would normally be, and is the CO cat. The pre cat lambda is at the input to this first cat, and is used to set fuel/air ratio.
The second cat is towards the rear, and is the NOx, (Nitrous Oxide), cat. It has two lambdas, one measures the efficiency of the first CO cat, (probably triggering the P0420), the other measures the efficiency of the NOx cat in removing the very nasty NOx emissions.
You should try to find out what the custom exhaust consists of, because, unless it contains the original two cats, and the three original lambda sensor, then it's likely, that emissions may not meet the MoT requirements. The "Catalyst System Efficiency Below Threshold" DTC, (P0420), hints in that direction.
If it were me, I'd want to establish the extent of the exhaust modifications, any deviation from the original functionality, and their implications.
As
@Andrew says, there is also a problem with the inlet flaps not moving through the required arc, or maybe not moving at all, (P1031), which has it very own, (and very long), thread. Careful diagnosis is advisable, before deciding on the fix.
You would not be the first to not worry about the EML, (although some have, subsequently, regretted it), as the car runs well. Again,if it were me, I'd want to know what is going on, before deciding not to worry. Or get it fixed, of course ...
If you bought from a bona fide trader, don't delay in informing them that you consider that car to have serious "pre-existing faults" as you may be able to recoup the cost of repairs.
Mac.
Edit: The NOx sensor, in the second cat, is not a simple lambda.