This is the 5555th most frequent Korean word.
못했지만
"Couldn't but" or "Didn't do but."
The word '못했지만' indicates that an action ('운동을 못했지만') was not accomplished in the past, but a contrasting situation follows ('오늘은 시간이 있어서 할 수 있어요').
어제는 바빠서 운동을 못했지만, 오늘은 시간이 있어서 할 수 있어요.
Yesterday, I couldn't exercise because I was busy, but today I have time, so I can do it.
The word '못했지만' is used to convey that the speaker was unable to do something (prepare for the exam) and is contrasting this with their current hope or wish.
시험 준비는 못했지만, 좋은 결과가 나오길 바랍니다.
I couldn't prepare for the exam, but I hope for a good result.
Here '못했지만' reflects the lack of ability or opportunity to perform an action ('prepare a gift') and provides a hopeful or positive rejoinder ('즐거운 생일을 보내길 바래요').