hiccup has 2 meanings:
One meaning is しゃっくり; or ‘an interruption in our breathing’. One hiccup is just one interruption in our breathing. But if you drink coca-cola very quickly, for example, you might hiccup again and again for a long time. In that case, it’s natural to say ‘I’ve got the hiccups’.
A hiccup can also mean a small problem; an interruption in something working normally. For example, ‘a hiccup in our plans’ could mean ‘a small delay in our plans’.
A: I ate too much. I’ve got the hiccups!
B: Have you tried holding your breath? Or shall I try to scare you?
A: We are trying to use new Conferencing Software, so of course there will be some hiccups.
What do you do when you get the hiccups?