Q: Is it permissible to make extra duas in prayer? Like when prostrating or when finished with the tashahud before saying the salam.
A: It is permissible to make extra dua in prayer in both places with duas that have been reported in the Quran, Sunnah, from the companions, and pious predecessors.
It is also permissible to make supplications for something of the Hereafter even if they do not resemble the supplications mentioned above.
Making dua for worldly delights like a big house, new job, pretty spouse will invalidate the prayer.
Making extra dua using the kaf al-khitab (May Allah forgive "you" father) will also invalidate the prayer.
Source: Sharh al-Muntaha