Devyani (Naman's mother, Mishti's dadi) gave away Mishti to Rajshri to take care of because Misti's mother remarried and Naman the father was in jail. She wanted Mishti to grow up in a loving environment that Rajshri Maheshwari (Akshara's mother) could provide. She raised both Kuhu and Mishti.
This was explained last week on YRKKH where Kuhu, Mishti, and Rajshri made a guest appearance.