That looks quite different from the cable adjuster on mine, it actually looks a bit like a brake/clutch cable adjuster from something else. Either way I don't think that's the root of your problem - The only purpose it serves is to set the amount of free play in the cable (could do with a locknut to keep it in place, probably what the tape was for.).
I can't really explain why it would leak petrol from there as fuel should not get that far up unless it's a really loose fit & is allowing it to blow past. (this would make the bike harder to start too as air would be able to get sucked past on induction)
If I was you then I'd just find & install a suitable locknut, adjust the cable till there's about 1mm of slack & wrap some more tape round. If the bike still seems hard to start then a good carb strip & clean should show up any other issues there may be, starting with the cold start piston & it's rubber seal as this would cause flooding if it was leaking.
P.S. Actually, while you're adjusting the throttle cable make sure the cold start also has a little slack when turned off in case the cable's holding the piston slightly shy of its seat.
Hope it's nothing too serious as it's a lovely day to go for a ride. Oh, and welcome to the forum.