Yes the TPS can affect how it runs to a point , but judging by your valve specs , I am going to guess that this is the issue , 0.004 would be a minimum I would be comfortable with , and some actually set them at that , but anything below that and you will start getting the popping and hard starting issues your experiencing .... On a related note , once the valves are in need of adjustment like that , they are on their way out and can last 1 or 2 rides , or 1 or 2 months , but they will zero out at some point soon , so you will need to do a valve job , if he is not going to be racing , I would recommend Stainless Intakes to replace the Titanium if your looking for longevity and less maintenance , they won't really affect performance as long as they are matched with springs to compensate for the extra weight of the Stainless valve
I run them in my 450R and have raced it with no issues , and I ride it pretty much every weekend at the track , I installed the Stainless 3 years ago and have not had to adjust them since , I also have them in my 450X and installed them in 06 and have not had to adjust them since , you can go with Titanium if you wish , as they have new materials that last much longer than OEM did back then ......That part is up to you , but just be warned that once the valves move , time is minimal before they are done
Back to subject:
Adjust the valves , and if you still have popping and starting issues , I can give you all the specs and post testing page images for you to follow directly from the Factory Honda Service Manual if need be to confirm the TPS is working properly and in the correct position .... but if you are planning on keeping this bike , I would recommend getting a Factory Service Manual since you seem to be mechanically inclined enough , you would benefit greatly from having it ....and if you still have questions , this is the place to ask
One other thing , sometimes just cleaning the jets does not clear them , mainly the Pilot jet , I would recommend just replacing the Pilot , as they are only about $6.00 or so, and that way you know it is clear , it does not take much restriction in them to give the issues your experiencing .....Though I think the valve lash is the primary issue since you gave that info ....... Report back with results from the valve adjustment and we can go from there ...