Welcome!
well..
this is kinda backwards to how questions around here usually come up,
but no worries, we can tame this horse either way
single seater live axle with a kid on board.
14" of rear wheel diameter.
Now we would need to know how fast you would want your kid to go roughly.
depending on age let's just assume 20'ish mph max speed for now.
Since all typical industrial engines are governed to run at 3600 rpms,
we now can calculate the gear ratio to be about'ish 7:1 (seven times as many teeth on the axle as are on the clutch.. 10-70 or 12-84 )
that'd result in 21.42 mph (close enough for a first guess

)
In order to determine the
let's just assume we talk about 220 lbs or less here (kart WITH kid)
In order to do that reliably with a standard cent clutch you'll need ~6hp
(a good 5.5hp engine will do.. frankly I'm fairly certain a briggs 5hp flathead should do as well.. nominally 6hp struggle less in case the kart's meant for off road backyards and uneven terrain)
The lighter the kart, the less hp you need of course, and the faster you want your kid to go, the more hp is needed.
If you add a torque converter instead of a standard centrifugal clutch,
values change slightly.
Limit wise: 5hp and 6.5 ratio for a 250lbs kart/driver should work with a high quality clutch (chinese cheap ones may fail due to overheating)
slower isn't an issue as long as the sprocket don't get too big to eventually touch the ground (in whcih case you'd need a jackshaft and compound gearing)
So, let us know about some more details and we can give a more exact value.
first estimate of 220lbs and 21 mph.. 6hp engine with 7:1 gear ratio.
'sid