Also of note is that 손목시계 is actually, more accurately, "wristwatch" instead of simply "watch". I don't know if "wristwatch" isn't accepted as an answer or not but that I found that good to point out

saying 'putting a watch on with one hand' is more colloquial and natural than 'putting on a watch with one hand'

"wristwatch" is not accepted, strangely. I flagged it.

Nah, putting on a watch with one hand is near impossible, not just difficult.

Wristwatch is still not accepted. And 손목 literally means wrist...