Hello. I thik when we want to ask or request someone for a thing, or doing something, say"kudasai." Like Kore(this) wo(a casae particle which shows a object of actions.) kudasai. And as you said, "Mite kudasai." And it's also used the case like beg a pearson's pardon for our actions. For example, "Ikasete kudasai." (Let me go.) I wonder if you see with my poor explanation. Anyway good luck!