What is the intended meaning of Fajr prayer, Dhuha prayer and Subh prayer and what is the difference between them? Is there a supplication in Fajr prayer like that of Witr prayer?

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

Fajr prayer and Subh prayer are two names for the same prayer which is one of the five prayers that Allaah imposed upon His slaves. It consists of two Rak‘ahs featuring audible recitation of verses from the Quran. Its time starts from the break of dawn and this is its best virtuous time. Its optional time extends to the appearance of the white light and its specified time ends at sunrise.

The Shaafi‘i and Maaliki schools of Fiqh believe that making Qunoot in Fajr prayer is permissible. According to Ash-Shaafi‘i school, Qunoot is made after rising from the second bowing. For the Maalikis, it is made after recitation and before bowing. If one forgets it and bows, he may make Qunoot after rising from bowing. They cited the Hadeeth in which Anas, may Allaah Be Pleased with him, narrated that the Prophet, sallallaahu ‘alayhi wa sallam, "continued to make Qunoot in Fajr prayer till he left this world." [Al-Bayhaqi, ‘Abdur-Razzaaq and Ahmad]

Others view that making Qunoot in Fajr prayer is not permissible because the Hadeeth about this is not authentic from their point of view.

Dhuha prayer consists of two Rak‘ahs. It is permissible to  perform it as four, six or eight Rak‘ahs which is better. Offering Tasleem after each two Rak‘ahs is recommended. Its time starts when the sun rises above the horizon about the height of a spear until shortly before Zawaal (when the sun moves away from its central meridian). It is an act of the Sunnah. It is the prayer of the oft-repenting about which the Prophet, sallallaahu ‘alayhi wa sallam, said: "The prayer of the oft-repenting (i.e. Dhuha prayer) is offered when the hoofs of the young weaned camels burn from the heat of the sand." [Muslim]

Allaah Knows best.