How does a man make up for the Tarawih prayer that he performed incorrectly in Ramadan, knowing that he will make it up after Ramadan, and does he make it up with the intention of it being Tarawih?

Whoever misses the Tarawih prayer is not obligated to make it up. The Hanafis and Hanbalis are of the view that it is not to be made up because it is not more emphasized than the Sunnah prayers of Maghrib and Isha, which are not made up. The Hanafis believe that making it up would be a supererogatory (nafl) act, which is desirable but not Tarawih. Conversely, the Shafi'is hold that it is recommended to make up a time-bound supererogatory prayer. The Malikis believe that supererogatory prayers are not made up, except for the two rak'ahs of Fajr. So, whoever wishes to make it up should intend to make up the Tarawih prayer.