Tarawih Prayer is a confirmed Sunnah that the Prophet (peace and blessings be upon him) performed. It is greatly rewarded by Allah the Almighty, the Messenger of Allah (peace and blessings of Allah be upon him) said, “Whoever observes optional prayer at night during Ramadan out of faith and in the hope of reward, his previous sins will be forgiven.” (Related by Al-Bukhari and Muslim) Muslims are keen to observe Qunut during the month of Ramadan in hope of answering their supplications.

The Messenger of Allah (peace be upon him) said, “There are three whose du`a’ is never rejected: a just ruler, a fasting person until he breaks the fast, and the prayer of one who has been wronged.” (Related
by Ahmad) Qunut, according to the definition of the jurists, is ‘a special du`a‘ (supplication) offered during prayer at a specific point while standing.” It is prescribed in Witr prayer after the ruku` (bowing), according to the more correct of the two scholarly opinions. 

In his answer to your question, Dr. Hatem Al-Haj said:

“Qunut is to be established in witr. However, scholars disagree over doing it aloud in the tarawih. The ‎Shafi`is and Hanbalis allow this; some throughout the ‎month, and some during the latter half, and that is what has ‎been authentically reported from the righteous ‎predecessors. ‎
The supplication in this qunut is often protracted by ‎the imams, and they tend to excel in beautifying their ‎voices with it and give too much attention to the ‎rhymes. Most of them make their own supplications, which are ‎not infrequently mistaken. The advice to the imams is to ‎limit themselves to the reported supplications in the Qur’an ‎and Sunnah to avoid making mistakes. If they needed to ‎use other supplications, they should not exaggerate and it ‎may be warranted to show them to a scholar first.
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As for the witr, Imam Ibn Taimiyah said that there are ‎three modes of witr, in ‎accordance with the various positions of the followed ‎imams:

1- Three rak’ahs, like ‎Maghrib, as done by the ‎Hanafis.‎‎

2– One rak’ah done after Shaf ‘.

‎‎3 – Three with ‎one tashahhud at the end. He indicated that the second and ‎third are correct, but went on to say that if the ‎congregation wants it done like Maghrib, the imam may ‎yield for the conciliation of hearts, and since this is a ‎position of a great and followed imam. ‎ This is an indication of the value of conciliation and ‎unity.”