The first thing to do is to be very clear what you *legal* rights are in this situation. What are your rights when it comes to claiming sick leave? Is your employer required to provide "reasonable accommodation" for health issues or disabilities? Does this apply to your case? What sort of documentation do you need to get this, if applicable, and do you have it? If you are scheduled for a shift, and don't show up due to a doctor's appointment, are they legally allowed to fire you?
This is very important, because it determines how you deal with your boss/HR. If you are asking for something that they are legally required to give you, you go in will full documentation, explaining what you need, in a polite, reasonable fashion, matter of factly.
If you are asking for something that they *aren't* legally required to give you, then you're going in as a supplicant, and you need to have a strong a position as possible - definitely be polite, because you're asking them to do you a favour. Have a plan for how you can balance things - a willingness to work unpopular shifts, last minute shift or double shifts to make up for time lost due to appointments, for example. Do your best to be clear about logistics - how often do you have doctor's appointments, how much notice do you get, is having to leave in the middle of a shift due to an emergency a regular or rare occurrence?
You want to project an impression of an eager, competent employee who is trying to do their job well, but needs some (minor) accommodation to do that. If you come across as demanding, annoyed, or entitled, it will be a lot less likely that they'll help you.
If they aren't required to accommodate your needs, and don't want to, then it's hard decision time. You need to think seriously about what your options are if you can't manage to balance your job and your health. This is a possibility - part time and casual workers in low paying jobs tend not to share a lot of the legal protections accorded to other types of employees, and don't have the ability to tell their boss to shove it and stay employed, particularly in today's economy.