Ok, so you will all know I'm spinning my rubber and getting nowhere fast.
I'm just about to go see A4Audi for a diagnostic - they have already quoted £846 for a new haldex controller - OUCH!
In the meantime I did some noseying around and found this excellent lead from the Audi TT Forum.
My apologies to username bhoy78 whose text I have nicked but I didn't want to make anyone feel inadequate by registering my RS3 on the TT forum!
Also see Auto Fault Finder here : http://www.autofaultfinder.com/
I had the n373 fault which autofault finder fixed. Despite it showing the fault n373 which as you say indicates that the board is likely suffering from corrosion that I was told it is much more likely to still be the pump. In my case it wasn't the pump and was just corrosion on the board which was fixed by cleaning and resealing. If you have the board out check that nothing on the board needs a re solder as I have seen someone who's fault was caused by this problem in the past.
You can also use vcds to check the measuring blocks, in my case I didn't have the correct voltage.
I really don't know a lot about this stuff but from what I read and was told by auto fault finder that the pump is much more likely to go. The oil filter at the pump gets clogged up and causes the unit to burn out. Audi do not sell this filter separately but the same part is used and is classed as a serviceable item across the rest of the gen 4 haldex vag range.
To buy the filter it is around £70/80. Pretty sure the guy told me they and awesome gti can supply them.
Auto fault finder will also fix your pump but don't think they do that by post although it maybe worth asking. For a drive in/out pump repair they charge £300 which is £80 for the filter, £80 for the haldex oil change so I can only assume that they must repair other components on the pump at the same time.
If your close enough I would say it's worth taking your car in, they told me they can fix 95% of haldex units regardless of the problem. That the have lists of the corresponding part numbers, have other haldex units that they can take parts from if required.
For reference mine was the n373 board, stripped cleaned and resealed for a drive in service cost £160 it has been perfect since and has no vcds errors.
I am assuming your car is same as mine with a gen 4 haldex, I think some TT's earlier ones have a gen 2 unit.
edit just cut and paste this from their fb site, it wasn't my car they were referring to a lady who the main dealer had tried to take for 2.5k
We are seeing more and more late model low mileage VW/Audi vehicles with Haldex issues. These generation 4 Haldex vehicles are burning out the pump due to clogged filter units but are unavailible to buy from VW/Audi/TPS as they are "fitted for life" and a "non service component". We find this shocking for 5 year old 50k mile vehicles, but not as shocked as the owners who are told they need a complete rear Haldex coupling at the best part of £2500!! We stock replacement filters, oil and pumps to revive these non functioning Haldex systems. As such we offer a same day drive in drive out repair service with no delay on parts orders. Also we can supply filters mail order.