gustapus

I have a 2001 TT Mk.1 convertible/roadster with the 1.8L engine. On a cold start, when the secondary air injection (SAI) pump kicks in, the engine idle increases (as it should) but the engine starts to idle roughly, with one or the other (usually #3 or #4) cylinders misfiring. As soon as the SAI pump kicks off, the engine idles perfectly. Sometimes it sets a Check Engine Light (CEL) with a cylinder mis-fire code, sometimes not, depending on how long the cold/SAI idle continues. It had the ignition coil campaign performed a number of years ago. The SAI associated tubing has been replaced/checked recently by an Audi mechanic. I have a MAF sensor in transit, and replaced the (apparently) never-changed-since-new fuel filter. I have new coils also in transit. But I fear I'm missing something and just "replacing with known good components". Oxygen sensor, maybe? Which one? If anybody else has had this problem, I would appreciate some insight. Help, please!