Is it permissible to raise one's hands in supplication for a specific need after bowing (Ruku') in the last Rak'ah of Duha prayer and Qiyam al-Layl (night prayer)?

It is permissible for you to ask Allah Almighty for all your needs during prayer, whether it is an obligatory (fard) or supererogatory (nafl) prayer. This should be done in the positions of supplication within the prayer, such as during prostration (sujud) and between the tashahhud and the taslim. As for supplicating with raised hands after bowing (ruku'), it is not prescribed except in cases of Qunut.