A lot of such common phrases are tied to some specific background or history. People typically say 추가요 in informal diners, like "밥 둘 추가요" (Two more rice bowls please"). This phrase tends to be shouted directly to the counter (usually not asked first), so you don't want to be lengthy. And since it's at a place you order things, 추가요 is as clear (maybe clearer) as equivalent phrases like 추가해요. (BTW, all 헤's in your post are wrong - they should be 해). It is essentially the single most effective word in that context. An interesting aside is that 추가요 has caught on, so that some start to use it in completely different contexts. For example, I've seen several cases where people use it to mean "Here's another so-and-so", in a derogatory sense. Here's a scenario on an Internet site, where the comments go like this: AA: (makes a point about something) BB: (apparently ticked off by AA's comment) 여기 뭘 모르는 얼간이 하나 추가요! (Here's another clueless fool!) He's saying the other guy is a fool, with the added effect he's no more than a cheap additional dish you shout out. So 추가요 seems to be even more popular now :-)

추가 is a noun which means "addition" 추가하다 is a verb which means "to add" 요 can be used after a noun. 운동요 or 사랑요 or 커피요 is correct sentence. 너는 무엇을 좋아해? 운동(요). It is contracted form of 운동을 좋아해(요) Just same in English What would you like to drink?(무엇을 마시고 싶어?) Coffee.(커피요) = I like to drink coffee(커피를 마시고 싶어요) 무엇을 원해요?(What do you want?) is omitted in this context. 샷 추가요(extra shot) = 샷 추가를 원해요(I want an extra shot)