If he missed something at the beginning of Taraweeh, how can he make it up?
If you miss ‘Isha’ and you arrive when the imam is praying Taraweeh, it is better for you to join him with the intention of praying ‘Isha’. When he says the salaam, you should complete your prayer and not pray alone or with another jamaa’ah (congregation), so that there will not be two jamaa’ahs at the same time, which may lead to distraction due to the intermingling of voices.
As for what you have missed of Taraweeh, if you wish you may do it, so you may pray two rak’ahs when the imam prays Witr, then pray what you missed, then pray Witr.
What is meant by praying two rak’ahs when the imam prays Witr is that you do not say salaam with him when he ends his Witr, rather you should stand up and pray another rak’ah, then say the salaam.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) was asked: If I join the people when they are praying Taraweeh and I have missed part of it, should I make up what I have missed after Witr or what should I do?
He replied: Do not make up what you have missed after Witr, rather if you want to make up what you have missed, then pray two rak’ahs when the imam prays Witr, then offer the prayers you missed, then pray Witr. Here there is an issue to which attention should be drawn: If you arrive when the imam is praying Taraweeh and you did not pray ‘Isha’ yet, what should you do? Should you pray ‘Isha’ on your own or join the imam in Taraweeh with the intention of praying ‘Isha’?
The answer is that you should join the imam in Taraweeh with the intention of praying ‘Isha’. When the imam says salaam from Taraweeh, then stand up and make up the rest of ‘Isha’. Imam Ahmad (may Allaah have mercy on him) mentioned the exact same issue, and this was also the view favoured by Shaykh al-Islam Ibn Taymiyah, and this is the correct view, because the correct view is that it is permissible to offer an obligatory prayer in congregation behind an imam who is offering a naafil prayer, based on the evidence of the hadeeth of Mu’aadh ibn Jabal (may Allaah be pleased with him) who said that he used to pray ‘Isha’ with the Prophet (peace and blessings of Allaah be upon him), then he would go back to his people and lead them in offering that prayer – it was naafil for him and obligatory for them. End quote from al-Liqa’ al-Shahri.
It is better for you to make it up in congregation if you can, but if that is not possible then there is nothing wrong with you making it up on your own.
And Allaah knows best.