"May I try the hat on?"
Translation:Можна поміряти шапку?
Чи: thanks, wasn't accounted for, added
Приміряти: thanks again, the version with "при" was not added
Цю: I think it's not accepted because there is no "this" in the English sentence
Шляпа: in the dictionary it's marked as spoken, colloquial; the proper word would be шапка or капелюх (also added)
"Чи" literally means "or" or "whether". It adds politeness and makes the question not so abrupt (compared to "Я можу приміряти цю шапку?"), especially if it's the first thing you say to a person and not a part of a long conversation :)
Kind of similar to the effect of using conditional tense in English, I guess? Like using "could" and "would" in a sentence to sound polite.