I assume you mean Autumn Ring Mini? That is a pain in the a$$ one (I struggled with that one on GT5). Some people say just be on the throttle the whole lap with a slight lift at turn 1 but I am a bit heavy handed for that. For me, I did the following:
Turn 1: Stay on the outside, Brake late and hard until you reach the apex then just let the kart coast until straight again and then accelerate (I usually overtake 3 or 4 karts on this corner);
Turn 2: Same again, brake late, coast till straight then accelerate,
Turns 3 and 4: Keep on full throttle, stay off the curbs
Turn 5: Lift off and coast till past the apex and then accelerate
Turns 6,7 and 8: These are all full acceleration with no lift or braking.
When I managed all the laps with no mistakes, I overtook the leader just after turn 4 on the second lap. The main thing is to stay off the curbs as much as possible, each time you try just push as hard as possible and if you spin then you know to back off slightly at that point next time round. Some people can do the whole track almost flat out, others, like myself (no wheel, only a DS3) need to lift off the throttle at certain points. Maybe reduce the controller sensitivity in the options menu. And, finally, don't just aim for the win each time, just aim to improve your time or just improve one corner at a time. If you can perfect one corner on one run, another corner on another and then string them together you will find it all comes together. Hope that helps a bit and good luck