How to use "top me off"
The whole sentence is "miss, could you top me off" and it's said to a bartender. It seems to mean " can I have more or can I get a refill " . But what does it mean exactly?

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To "top off" means to fill some container until it is full.  There may be some liquid already in that container.  In the case of a drink, it means add enough liquid to the glass I am drinking from to make it full, even if there is already some in the glass.  If you wanted another drink, you would typically say "can I have another (please)".

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