Assalaamu alaykum. With regards to the night prayer in Hanafi fiqh:
A. What is the method of making supplication during prayer? In which position? Can one also supplicate in his own language? Please elaborate.
B. Does the night prayer time end when the true dawn occurs or when the azan for the Fajr is called?
May Allaah reward you.

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 is His slave and Messenger. 

After investigation in the books of the Hanafi School that we have at hand, we have not come across any special supplication or any specific position where to say the supplication in the night prayer according to the Hanafi School. The night prayer is like all other prayers; it is permissible to supplicate as much as possible while prostrating.

Although the Hanafi Scholars said that the worshiper does not say other than tasbeeh (praising Allaah by saying Subhaana rabiya al-A’la) while prostrating, they stated that thikr (remembrance of Allaah) in the prostration other than the tasbeeh stated in the Sunnah is interpreted to apply to voluntary prayers (not obligatory prayer) such as the night prayer.

Al-Bahr Ar-Raa’iq reads, “A praying person, when bowing down or when in the prostration, does not say other than the legislated tasbeeh and other matters mentioned in the Sunnah; this is interpreted to apply to the voluntary prayers, whether it is the night prayer or any other.”

The correct view is that it is permissible to say other than the tasbeeh, such as supplicating, even in obligatory prayers because of the general saying of the Prophet “Supplicate much while prostrating as it is most likely that your supplication will be answered (in that position).” [Muslim]

As regards supplicating in one’s own language, if you mean in a language other than Arabic, then the scholars differed in opinion in regard to the ruling of supplicating in other than Arabic language inside the prayer. Some of them held that this is not permissible, and another group held that this is permissible.

Some others were more specific: they distinguished between someone who is capable to supplicate in Arabic and someone who is incapable.

There is no doubt that it is better for those who can supplicate in Arabic while understanding the meaning to supplicate in Arabic. If one is not able to do this, then he may supplicate in his own language.

The Fiqh Encyclopedia reads, about supplicating in a language other than Arabic:

“Supplicating in a language other than Arabic in the prayer:

1- What is reported from the Hanafi School in regard to supplicating in a language other than Arabic is that it is disliked; because ‘Umar forbade speaking in a language other than Arabic while in prayer. The apparent reasoning is that supplicating in a language other than Arabic is khilaaf al-awla (i.e. contradicting that which is better) and that the dislike of doing so is for close to haraam (prohibition), but it is likely that supplicating in a language other than Arabic is not only makrooh (disliked) for tahreem (close to haraam, i.e. strictly abominable) in the prayer, while it is makrooh for tanzeeh (close to halaal [permissibility], i.e. religiously permitted) outside the prayer.

2 - The Maaliki School hold the view that it is forbidden to supplicate in a language other than Arabic – as quoted by Ibn ‘Aabideen from al-Qaraafi – giving the reason that this includes what contradicts glorification. Al-Laqqaani restricted the statement of Al-Qaraafi to non-Arabic that is not understood. However, if the person understands the meaning of what he is saying, then this is absolutely permissible whether inside or outside the prayer; as Allaah says (what means): {And He taught Aadam [Adam] the names – all of them.} [Quran 2:31] Allaah also says (what means): {And We did not send any messenger except [speaking] in the language of his people...} [Quran 14:4] This is also what Ad-Dusooqi has stated.

3- The Shaafi'i School elaborated on the issue and said: supplication in the prayer is either confirmed from the Prophet or not confirmed from him. The confirmed supplication is of three aspects:

- The most correct view, which is also the view held by the Hanbali School, is that it is permissible to supplicate in a language other than Arabic for whoever is not able to supplicate in Arabic, whereas this is not permissible for whoever is able to supplicate in Arabic because if the latter does so, his prayer is invalid.

- The second view: it is permissible for whoever speaks Arabic well and for others.

- The third view: It is not permissible for either of them because there is no need for it.

As regards the supplication that is not confirmed from the Prophet in the prayer, it is not permissible to say it or invent it in a language other than Arabic; there is a consensus about this...” [End of quote]

Concerning the night prayer, it starts after the Isha prayer and lasts until the Fajr starts, pursuant to the hadeeth by Ibn ‘Umar in which the Prophet said, “If the Fajr starts, then all night prayers and the Witr are finished; so perform the Witr prayer before the Fajr starts.” [at-Tirmithi]

So what is taken into account is the true dawn, and not when the athaan is called by the muaththin because the muaththin may call the athaan before the true dawn or after it.

Allaah knows best.