Whenever you buy a used bike, you are always left to wonder what "improvement" the PO(s) did (if it worked at all or made things worse) until you get into the bike and figure things out for yourself.
These bikes are really pretty easy to work on if you are even half-way handy and take your time.
Making sure you have all the airbox in place and there are no cracks or leaks in the snorkels, baffles, etc, is a good first step and takes almost no effort.
I doubt an air filter makes much difference unless it is just full of garbage, but if you have made it far enough to pull the airbox and verified all the airbox(s)/snorkle integrity, pulling the carbs is almost as easy. I would not waste my money on a K&N filter, probably the T3 unit can be easily cleaned and is probably a better filter to begin with. It is interesting if you run these bikes with the airbox off completely (i.e. testing or something), they can barely get out of their own way and stumble terribly.
Pulling the carb rack is a little exercise and the discussion of how to do that for another thread (you probably have Keihins? Guessing, they changed over from Mikunis at some point). There are some tricks to making that easier, feel free to send a PM. No need to separate the carbs, the rack of three goes in and out in one piece.
The next step might be checking the pilot jets are not plugged (which is very common and very quick to happen if the bike has sat for any length of time) and that everything in the carbs is at least set-up right. This will also give you an opportunity to verify exactly what needles and main/pilot jet sizes are in place, no cracked rubbers, float heights. Don't even have to buy anything. If the EPA caps are in place on the bottom of the carb, remove them and set the idle mixture screws to 2.5 turns out to approximate the Euro settings they left the factory with.
Stock/OEM settings with everything working right and they are sweet running machines for everyday use.
Pulling the carb rack and peeking inside them sounds daunting but it is really pretty easy if you take your time - a carb only has maybe a dozen parts in total. If you want go ahead and replace the few rubber o-rings and float needle and other rubber bits for what is considered a complete rebuild.
And if you've done that, sync the carbs upon final assembly if you can beg/borrow a morgan carbtune or similar - well worth the 10 min it takes.
Do all this and you'll likely save yourself $500+ that the stealer will charge you (at least once, maybe twice) and have some answers instead of guesses, and likely end up with a smooth running bike you know well.
Of course you may stumble upon another fix that works for you, but I'd still pull the carb rack do the cleaning and rebuild as it will be necessary eventually and usually saves heartache down the road.
Of course there is always the potential to replace plugs (easy with the correct socket/tool, or maybe a bad coil, maybe spark plug wires, but I would leave the last two as this is less likely. Plugs are cheap and can't hurt (unless you are a klutz and bugger a thread up).
Tip - grab the haynes manual and the T3 Triumph Service manual if you are going to wrench or debug, both are good and cheap compared to some other makes.
