In general:
- during touch & go, you would retract flaps then apply full power
Your first reference for this answer would be the aircraft flight manual. I can't recall instruction for touch and go procedure in a flight manual I have seen, it is generally not something the aircraft manufacturer might want to instruct pilots to do.
Any flight manual, and all good airmanship, will direct that that the pilot "fly the plane first". This means, in order of importance: Maintain pitch control, maintain directional control, and maintain flying speed. After that, everything else is secondary. Anything which could interfere with the primary objectives must be relegated to lesser importance - this includes fiddling with the flaps. I have certainly experienced other pilots insisting that flaps not be retracted until 400 feet up after takeoff, so why would pilots be willing to reposition them on the roll, if there is the least risk of distraction?
The maintain pitch control means that whether you have power or not, prevent a pitch attitude which will result in an undesired stall. That is always most important. Directional control, go toward the point you intend (runway centerline, in this case). Maintain flying speed can be associated with the use of power (assuming that you are managing pitch well already). If you need to fly, and climb, manage the power to accomplish this. I agree that flaps play a role in flying speed and pitch, but it is secondary, knowing that the aircraft has demonstrated an adequate climb with full flaps extended. If anything else you do at the controls could interfere with the prime objectives, don't do it! To me, a configuration change on the roll, as happens in a touch and go, increases the risk of a loss of directional control, so must not be done, if directional control cannot be safely maintained. If this means that you come airborne with full flaps while aborting a landing (you could be on the runway already when deciding to abort), so be it.
If you have decided that a full stop landing is no longer what you want to do, you've decided to abort/overshoot/go around. This can be done after touchdown, as maybe something enters the runway ahead of you. If that happens, you're not going to roll toward whatever it is, while looking for the flap switch, before adding power, you're just going to open the throttle and go, getting the flaps as best you can as you do. The aircraft has demonstrated that it can do this without requiring unusual pilot skill
Touch and goes require at the very minimum, a mindset change on the roll, though more likely, a configuration change as well. I would expect huge philosophical resistance to touch and goes in retractable aircraft, where flaps are being retracted on the roll. So, flap equipped fixed gear planes probably deserve similar caution, just from the distraction and loss of directional control risk perspective.
Touch and goes become a pilot workload issue. If they can be accomplished safely, they are a time saver, but there is a fine line, where they are no longer safe, and the skill lies in making that determination - sometimes very quickly. This determination has to allow for a bit of unexpected after the decision is made - like a misbehaving flap switch.
I recall doing solo touch and goes in a C 180 float plane decades ago. After landing with 40 flap, I retracted them to 20 while on the step. Though precise directional control and runway dimensions were not a concern, sudden pitch changes while on the step a not good. The flaps jammed, and would not retract to less than 20. Rather that diagnose this while planing across the water with the full power I had already applied, I took off. The flaps could not be repositioned at all, so I flew home and landed with 20. It turned out that the flaps had jammed because of a broken flap track - that could have been very much worse! In hind sight, staying on the water would have had some merit in this case, though out in the hinterland is not the ideal place to shut down and look for repairs!
If you're going to do touch and goes, you have to assure that you have the excess pilot capacity to deal with things like flaps, and the extra runway space to assure it can be safely done.