This is a race in the Casino Palace. On normal, this should be like a usual 'standard class' race. Just use your boosts at the right times, and angle your jumps off of the jump pads correctly for good jumps. Try not to stay in the air too long or you'll lose speed and the other racers will pass you. Try your best to heal as much as you can because you are going to need the boosts.
Hard and Very Hard modes are like 'Expert' and 'Master' classes respectively. This means that if you have raced in these classes, it shouldn't feel much different than other races. In these difficulties, you'll need to go on an almost non-stop boosting spree if you want to win. Boosting isn't enough for this. Billy, the monkey in the yellow f-zero machine, will have just about infinite boosting and your chances of winning with him alive are slim to none. He usually dies on his own on one of the jumps if you're going fast enough but otherwise, it would be better off if you restarted if he didn't die. Another threat in this mission is Super Arrow (brown mask with feathers on it inside of a red machine). He won't really die on his own as much as Billy does but you can easily just ram into him to get the job done.
In the final stretch of the final lap, everyone will boost like there is absolutely no tomorrow so you should do the same but keep your machine under control or else someone will pass you in the last second. It's not uncommon for them to pass you in the last second so don't be surprised. Just keep practicing and memorizing and you'll beat without a problem.