It depends where you kart.
There are some conditions when new tyres are slower than old tyres.
There are some tracks (reputedly mainly in the south) which don't seem to wear tyres so much.
Drivers at Larkhall for instance claim that after 1 raceday the tyres are not only 'dead' but worn below the depth nubs, while at Dunkeswell the track is (or was) very kind to tyres.
It also depends on your experience, if you are not driving up to the limits of your tyres or you have set the kart up wrong or your technique is bogging down the tyres that you have, then new tyres aren't going to make that much difference.
In addition, at any track, you are likely to see a half second to a second improvement in times over the day. (Look at laptimes on racedays, the finals are likely to be quicker than the heats and it isn't down just to the heightened sense of competition).
But for a good driver, on back to back testing, there's likely to be an improvement using new tyres, if only because the driver relies on them gripping more predictably than when they are worn. Andy Potter, a good driver, claims 2/3 a second a lap.
The answer is to test it and see. Scrub the tyres in (three or four medium fast laps),then let them cool.
Do the test with the tyres (used and new) at the same temperature (ie don't go out with 'warmed' new tyres and cold used tyres, or vice versa) and be honest about the traffic or mistakes that you make.
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