I've cried listening to PTG's Beautiful, but those were more awestruck tears than sad ones.
Anyway, a few songs that never fail to make me cry/sad/depressed are:
Father by Demi Lovato
Show Me the Meaning of Being Lonely by the Backstreet Boys (I was also pretty hormonal as a tween/teen when it came out, so just imagine my tears then! lol)
Sign of the Times by Harry Styles
(okay, enough songs about death lol)
Summer Love by One Direction
Amnesia by 5 Seconds of Summer
hmm... yep, can't think of any K-pop ones!