When I make more than one cup of coffee on Shabbos, I just refill the same cup and pour it into the new cup and don’t clean out the old cup. Is that a problem?
According to the Ashkenazi minhag, if the water in the cup is still warm (note: the little bit of water will cool in the cup very quickly) then you can pour the hot water directly on to it, because ayn bishul achar bishul. If the water has already cooled down, then all you have to do is to pour out any excess water, but it does not have to be totally dried out. If the water inside the cup would be uncooked, then the cup should be dried out.
Igros Moshe O:CH 1-93, 4-73 bishul 19, MInchas Yitzchok 9:30-31,Bris Olam 34, Beer Moshe 6-110, Meohr Hashabbos 4, Mishne Halachos 6-87, Shevet Halevi 7-42(2).