Kpop music shows are both live and pre-recorded at the same time, like how old sitcoms/sketch shows were: live audience and live performance are recorded. This allows for multiple takes, but are still technically "live."

During the live broadcast, I think the MC parts in between performers are live. Then for the performances, the pre-recorded performance is broadcast, but the artists also reappear on the stage and casually lipsync-perform over the pre-recorded audio. Then the sound guys mix in the audience screams during the broadcast live with the pre-recorded video/audio. This also allows for the performers to do stage shenanigans for the benefit of the studio audience.
Then, if there's an award at the end of the show, (I think only Music Bank still does) that's done live, as is the encore performance by the winner. (which is usually cut off for time)

An example of Music Core's pre-recording process can be found here, from 2:48 on the first video to 2:17 on the second.
And here's footage broadcasted vs. performance during broadcast.

There was once a "scandal" over a singer being denied a second take, but as you can see, the scandal wasn't over the fact that they do pre-recordings and second takes, but just that she aired complaints during the broadcast.