Praise be to Allah, and may Allah’s peace and blessings be upon the Messenger of Allah, his family and his companions.
In response to your question, I say:
There is difference of opinion regarding this issue. Ahmad and a group of Hanafite, Malikite, Shafite and Hanbalite jusrits and others are of the opinion that it is permissible to supplicate. Ahmad was asked about completing the Quran during prayer and he said: if you finish reciting Surah Qul A’udhu birabinnas then you should raise your hands and supplicate before bowing down. When he was asked about his proof he said: I saw the people of Makkah doing it and Sufyan Ibn ‘Uyaynah did it with them. Al-Nawawi said that completing the Quran during prayer is better and for that, the Imams who do it (supplication) base it on completing the Quran.