Answered by: Maulana Yusuf Khan
Can Taraweeh be prayed in Tahajjud?
In the name of Allah, the Most Gracious, the Most Merciful
The number of rakaats for Taraweeh prayer is 20 with 10 salam’s in between, this is the majority view according to the ulama. 
In terms of time, the Taraweeh prayer is prayed after Isha and before fajr according to the most correct opinion. 
Taraweeh is a prayer which is unique to ramadan hence the prayer time is specific to in between Isha and Witr as mentioned previously, as opposed to tahajjud which is prayed after esha and in the final third of the night. This differentiation of the timings and specificities make each of them independent from each other.
The majority of the Hanafi ulama agree that the intention for taraweeh prayer is fulfilled with a general intention of salah. 
It is not possible to pray Taraweeh in tahajjud as they are different, one is recommended to pray tahajjud after he has completed his 20 rakaats of Taraweeh as not only it is a sunnah, it is one that only comes once a year in combination with the sacred nights of Ramadan.
Only Allah knows best
Written by Maulana Yusuf Khan
Checked and approved by Mufti Mohammed Tosir Miah
Darul Ifta Birmingham