I tend to agree with Dom that the kart doesn’t really look like it’s hopping. The only place you can see it hop is at 0:21, where you have a big lock-up under braking and really upset the kart on the entry. Before tuning the kart you need to isolate that and make sure it isn’t your driving that is causing the hopping. If you are less aggressive on the brakes in that corner and don’t get the kart sideways under lock-up, does it still hop?
Hopping usually comes from overloading the chassis and making it jack weight too aggressively. This can be a setup problem, but if you are driving the kart too aggressively into the braking zone like at 0:21, you can also make the kart hop. You’re loading the kart up sideways on entry, then you have to put a bunch of steering input in after braking to get it to rotate back the other direction, which flexes the chassis too much and causes it to rebound and hop.
It’s okay to be aggressive in a braking zone, but we want to make sure we get the chassis settled back down before we initiate turn-in, otherwise you are compounding a bunch of inputs and forces through the kart, and when you go to turn-in, the kart will be unstable.
First, try and be less aggressive in that brake zone and see if you can get it to settle down. If that doesn’t work and it still hops, then you can try to tune it out.
Since hopping comes from too much or too quick weight-jacking, the first thing that stands out to me is that you have full caster in the kart. The Tony Kart has a lot of front geometry to start, so there is rarely a need for full caster. I would reduce the front caster to start. If you have a neutral pill where you can go to half-caster, I’d do that first, but if you only have the stock pills, I would set them to full-neutral (both arrows on the caster pills pointing forward or backward). The full-caster on the front end is jacking a LOT of weight. Probably also why the kart feels twitchy and nervous on harder tires. When you put the harder tire on, the hopping disappears because you aren’t gripping up when the weight transfers, and instead you just overload the less grippy tire and cause it to slide immediately with all that front-end in the kart.
On the OTK chassis line, you go stiffer on the axle to combat hopping. Generally a softer axle will rebound and spring more, so it increases rear traction and can induce hopping. But you shouldn’t be looking at axle changes yet. The N axle is the correct axle for 95% of conditions and should be the starting point all the time. Tune all the other easier options first and if you can’t get it to stop hopping, then you can look at starting at a new baseline with a different axle. If you want to eliminate hopping you would go to a stiffer axle (H, HD, HH etc.)
Given that it looks like it’s only hopping in one corner and nowhere else, my guess is that this is driving related, specifically to your braking technique for that corner. Remember that the kart is a constant, it doesn’t change throughout the lap, so if there is an issue in one corner, it’s likely not the kart causing it. If it was, the issue would appear in multiple corners per lap.