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. TheShafi`is and Hanbalis allow this; some throughout themonth, and some during the latter half, and that is what hasbeen authentically reported from the righteouspredecessors.
The supplication in this qunut is often protracted by
the imams, and they tend to excel in beautifying theirvoices with it and give too much attention to therhymes. Most of them make their own supplications, which arenot infrequently mistaken. The advice to the imams is tolimit themselves to the reported supplications in the Qur’anand Sunnah to avoid making mistakes. If they needed touse other supplications, they should not exaggerate and itmay be warranted to show them to a scholar first.
As for the witr, Imam Ibn Taimiyah said that there are
three modes of witr, inaccordance with the various positions of the followedimams:

1- Three rak’ahs, likeMaghrib, as done by theHanafis.‎‎

2– One rak’ah done after Shaf ‘.

‎‎3 – Three withone tashahhud at the end. He indicated that the second andthird are correct, but went on to say that if thecongregation wants it done like Maghrib, the imam mayyield for the conciliation of hearts, and since this is aposition of a great and followed imam.This is an indication of the value of conciliation andunity.”