Identifying the material (as you seem aware of) is the first thing you want to do. Unfortunately I don't think you will find that here. I suspect it may be a form of ABS or PE (both very weldable), but as someone already suggested, your best bet is probably to start with a Honda dealer.
Though plastics seem to last forever and are very resilient, they do age and fatigue. This is especially noticeable in an environment of vibration and UV exposure (something your ATV sees a lot of I suspect). The material along the cracks probably won't make for a good bond, either through heat or chemical.
If it's just appearance, I don't know if it's really worth the effort. As it may come out worse than what you have now.
If it's about function, and replacements cost too much or just aren't available, you may want to consider a mechanical solution. Such as a strip or patch of plastic or aluminum attached with pop rivets using suitable backing washers. This will divert stresses to the connection points an will probably fail themselves in time. But it may be enough till the rest of the machine is in the same condition.
Sorry, not much help. But maybe it will give you some alternative ideas. Good luck.