"Song and dance"
The verb is 추다, but some Korean verbs like "to dance", "to sleep", "to dream", are weird in that they need an object, so you actually have to say "to dance a dance". That's how you get 춤추다, which is just a short version of 춤을 추다.
Basically, unless you specify the dance, like 탱고를 추다 (to dance tango), or 삼바를 추다 (to dance samba), you have to write it as 춤을 추다 (to dance a dance). It's just how the language rolls.