I carry the following:
2 x spare tubes
2 x tyre levers
1 x p*nct*r* repair kit
3 x CO2 canisters
1 x strong tweezers with a serrated grip at the gripping end
I would always replace the tube rather than try to repair at the roadside. I find the simplest way to find the offending sharp object is this. Find easily identifiable spot on righthand tyre rim, hold tyre in left hand, place fingers of right hand inside tyre touching the centre and righthand wall, slowly rotate tyre with left hand till you feel the thorn, nail etc. Turn tyre round so, lefthand side becomes the right and repeat. Use the tweezers to remove the sharp object - in my experience 99.9% certain to be a hawthorn thorn.
The best way to avoid the problem is to have the best quality tyres inflated to maximum pressure. I've had this policy with excellent, 100% effective, results for 4 years and 5 months.