All perfect praise be to Allah, The Lord of the Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
Making supplication during prostration and after the last Tashahhud is permissible. This is particularly, when supplicating ease for yourself (Questioner) and for all Muslims in this life and in the Hereafter. However, it is recommended to abide by the supplications mentioned in the Prophetic Sunnah.
Al-Imam An-Nawawi (May Allah have mercy on his soul) said (What means):" According to our Madhab (Shafie school of thought), supplication is permitted regarding worldly life's and the Hereafter's matters, such as saying : O Allah, give me a lawful provision, house, son and a kind wife or free someone from prison or destroy another…etc and this is the opinion of Malik, Al-Thory and Isaac. Nevertheless, Abu Hanifa and Ahmad stated that it is only permitted to make supplication with the ones mentioned in the Sunnah and the Quran, since the Prophet(PBUH) said (What means):" "It is not permissible to talk during Salat because it consists of glorifying Allah, declaring His Greatness as well as recitation of the Qur'an," [Muslim]. By analogy with answering greeting and responding to the one who sneezes, it is said: (May Allah have mercy on you).
However, the Shafie scholars based their opinion on the following Hadith: The Prophet (PBUH) said (What means):" exert yourself in supplication in prostration." According to this narration, the Prophet (PBUH) permitted supplicating in any matter without any sort of restriction because he (PBUH) has made supplications in many forms, showing that there is no harm in this. Besides, Ibn Masso'd (May Allah be pleased with him) reported, that the Prophet(PBUH) said (What means):" O Allah! Save Al-Walid bin Al-Walid; O Allah! Save the weak people among the believers; O Allah! Be hard on the Tribe of Mudar; O Allah! Inflict years of drought upon them like the years (of drought) of the Prophet (PBUH) Joseph." [Agreed upon], also he said (What means):" I curse the tribes of Ri'l, Dhakwan, and 'Usayya for they disobeyed Allah and His Messenger (PBUH). This is in addition to many other narrations mentioned in this regard. Yet, the tribes mentioned in this narration are Arabic ones.
Replying to the Hadith cited by Abu Hanifa and Ahmad that supplication isn`t part of peoples' talk and that answering greeting and responding to the one who sneezes is considered from peoples' talk since they are address individuals, unlike supplication which is directed to Allah. And Allah knows best.