Supplicating during prostration can be done with both narrated and non-narrated supplications
2-5-2017 | IslamWeb
Question:
Please advise me regarding some supplications that I have come accustomed to saying during prostrations; I do not know if they are allowed or not. These supplications include sayings like 'Rabbi-shrah li sadri wa yassir li amri…' [My Lord, expand my chest and make things easy for me…], 'Allahumma waffiqni lita‘atika' [O Allah! Make me succeed in obeying You], 'Allahumma-hfath binti fulanah' [O Allaah! Preserve my daughter, so-and-so], 'Allahumma-ghfir liwalidayya' [O Allah! Forigve my parents], 'Allahumma-ghfir lizawjati warhamha' [O Allah! Forgive my wife and have mercy on her], and 'Allahumma akrim nuzulaha' [O Allaah! Give her generous accommodation]
Answer:
All perfect praise be to Allah, the Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
Supplicating during prostration is mandoob [recommended] because it is then that the slave is closest to his Lord, as the Prophet, sallallaahu ‘alayhi wa sallam, said, "The closest the slave is to his Lord is while he is in prostration, so abundantly supplicate then." [Muslim]
It is allowed to supplicate therein and elsewhere of the places in which supplication is enacted in prayer, regardless of whether that supplication is from the Quran, like His Exalted saying on behalf of Moosa 
: {My Lord, expand for me my breast [with assurance]…} [Quran 20:25], or from a Prophetic Hadeeth, like his, sallallaahu ‘alayhi wa sallam, saying, “Subhanak-allahumma rabbana wa bihamdik, Allahumm-aghfir li.” [Glory to You, O Allah, our Lord, and with Your Praise! O Allah! Forgive me!] [Al-Bukhari and Muslim]
It is also permissible to use that which has not been explicitly narrated, like your saying 'Allahumma waffiqni lita‘atika' [O Allah! Make me succeed in obeying You], 'Allahumma-hfath binti wa zawjati' [O Allah! Preserve my daughter and my wife], and so forth, as long as there is no sin or severance of kinship. This is according to the majority of the scholars.
Allah knows best.