I gave up on her because I couldn't get her father's blessing on our relationship.
I gave up hope for her to mend her ways.
この場合、次のような言い方ができますよ。
① I gave up on her because I couldn't get her father's blessing on our relationship.
「彼女の父親の賛成が得られなかったので、彼女を諦めた」
② I gave up hope for her to mend her ways.
「彼女が行いを改める見込みを諦めた」
ご参考まで!