First: Regarding making supplications along with the dhikr (remembrance) after rising from rukū῾, we say: Several ḥadīths have been narrated, some of which mention only the dhikr (like “Rabbana lakal ḥamd, ḥamdan kathīran…”) as in the ḥadīth of ῾Alī in Ṣaḥīḥ Muslim, ῾Abdullāh ibn ῾Umar in Ṣaḥīḥ Al-Bukhārī, and Abū Sa῾īd in Ṣaḥīḥ Muslim.
There is also a narration in Ṣaḥīḥ Muslim from ῾Abdullāh ibn Abī Awfā that includes both dhikr and supplication, in which he said: “O Allāh, purify me with snow, hail, and cold water. O Allāh, purify me from sins and mistakes as a white garment is cleansed of dirt.”
Second: The worshiper should not raise their hands for supplication after rising from rukū῾. Raising the hands for supplication during prayer is only done for the qunūt supplication according to the Ḥanafī, Shāfi῾ī, and Ḥanbalī schools of Islamic thought.
Therefore, the worshiper should let their hands hang by their sides after rising from rukū while making the dhikr and supplication together. Fatwā issued by Dr. Khālid Naṣr