It is better when reciting in Taraweeh to complete the Qur’aan once. This may be understood from the reports in al-Saheehayn which state that Jibreel used to study the Qur'aan with the Prophet (peace and blessings of Allaah be upon him) in Ramadaan, and review it with him.
Shaykh Ibn Baaz said (15/325):
It may be understood from this that if the imam recites the entire Qur’aan to the congregation during Ramadaan, this is a kind of this studying together, because this lets them benefit from hearing the entire Qur’aan. Hence Imam Ahmad (may Allaah have mercy on him) used to like to complete the Qur’aan with those whom he led in prayer, and this is akin to what the salaf did, namely liking to hear the entire Qur’aan. But this does not mean that he should rush and not be deliberate in his recitation, and not seek to be focused and calm, rather seeking these things is better than seeking to complete it.
Majmoo’ Fataawa al-Shaykh Ibn Baaz, 11/331-333
It says in al-Mawsoo’ah al-Fiqhiyyah (27/148):
The Hanbalis and most of the Hanafi shaykhs – and it was also narrated by al-Hasan from Abu Haneefah – are of the view that the Sunnah is to complete the Qur’aan in Taraweeh prayer so that the people can hear the entire Qur’aan in that prayer. The Hanafis said: The Sunnah is to complete it once. So the imam should not forsake completing it because of the people’s laziness, rather he should recite ten verses or so in each rak’ah, and thus he will be able to complete it. (This is based on the assumption that he prays twenty rak’ahs each night.) And it was said that in each rak’ah thirty verses should be recited, because ‘Umar (may Allaah be pleased with him) enjoined that. In that case the Qur’aan can be completed three times in Ramadaan.
Al-Kaasaani said: What ‘Umar enjoined was by way of doing more of a good thing, which is to complete the Qur’aan more than once. This is what was suitable for their time. But in our time it is better for the imam to recite on the basis of the people’s situation; he should recite whatever will not put them off from joining the congregation, because increasing the size of the congregation is better than lengthening the recitation. End quote.
What al-Kaasaani said is good, and the imam should pay attention to the situation of the people behind him.
It is not permissible for the imam to put people off by making the prayer so long that it becomes difficult for them and to think that if he does not do that he has done badly! What he should do is to encourage the people to pray even if that is by making it shorter, so long as the prayer is complete.
It is better for the people to offer a short but complete prayer than not to pray at all.
Abu Dawood said: Ahmad ibn Hanbal was asked about a man who recited the Qur’aan twice in Ramadaan, leading the people in prayer. He said: In my view this depends on the people’s energy level, and whether there are workers among them.
Ibn Rajab al-Hanbali said: The words of Imam Ahmad indicate that attention should be paid to the state of the people with regard to recitation; he should not make it too hard for them. This view was also echoed by other fuqaha’ among the companions of Abu Haneefah and others.
Lataa’if al-Ma’aarif, p. 18
Shaykh ‘Abd al-‘Azeez ibn Baaz was asked:
What is your opinion about what some imams do, choosing a certain amount of Qur'aan to recite in each rak’ah each night?
I do not see anything wrong with that, because it depends on the imam’s own assessment of the situation. If he thinks it is better to recite more on some nights and in some rak’ahs, and that this will benefit those who are praying behind him, and he feels that he has the energy for that and he feels that he is enjoying the recitation and wants to recite more verses to benefit himself and others, then he may do so. If his voice is good and he started to enjoy the recitation and feels humble and focused and hopes that this will benefit himself and those who are praying behind him, then if he recites some extra verses in some rak’ahs or on some nights, we do not know of any reason why he should not do so. The matter is broad in scope, praise be to Allaah.
Fataawa al-Shaykh ‘Abd al-‘Azeez ibn Baaz, 11/335, 336
Shaykh ‘Abd al-‘Azeez ibn Baaz was also asked:
Should the imam pay attention to the situation of the weak such as the elderly etc when praying Taraweeh?
This is something which is required in all the prayers, in Taraweeh and in the obligatory prayers, because the Prophet (peace and blessings of Allaah be upon him) said: “When any one of you leads the people in prayer, let him make it short, because among them are the weak, the young and those who have needs.” So the imam should pay attention to the congregation and be kind to them with regard to praying qiyaam in Ramadaan and during the last ten nights. People are not all the same, rather they vary. So he should pay attention to their different situations and encourage them to come. When he makes the prayer too lengthy, he makes it difficult for them and puts them off from attending. He should pay attention to that which will encourage them to attend and pray, even if that is by making it short and not making it long. If the people concentrate during the prayer and find peace in it only for a short time, that is better than a long prayer in which there is no concentration and people only feel bored and tired.
Fataawa al-Shaykh ‘Abd al-‘Azeez ibn Baaz, 11/336, 337
In the answer to question no. 20043, we stated that reciting part of a soorah in prayer is permissible, but it is better to recite the soorah in full, because this is what the Prophet (peace and blessings of Allaah be upon him) usually did.
Some scholars – such as Ibn al-Salaah – made an exception in the case of Taraweeh prayer, and said that it is better to recite part of a soorah in them, so that (the imam) may achieve the Sunnah of completing the Qur'aan in these prayers.
It says in Tuhfat al-Muhtaaj Sharh al-Manhaaj (2/25):
From this it may be understood that if he wants to recite the whole Qur’aan in Taraweeh it is better to recite part of the soorahs (in each rak’ah), otherwise it is better to recite the whole soorah (in one rak’ah). End quote.
It says in al-Mawsoo’ah al-Fiqhiyyah, 33/49:
In one of the two reports narrated from him, Maalik regarded it as makrooh to recite only part of a soorah,
The Shaafa’is and Hanbalis are of the view that it is not makrooh to recite only part of a soorah, because of the general meaning of the verse in which Allaah says (interpretation of the meaning):
“So, recite you of the Qur’aan as much as may be easy for you”
And Ibn 'Abbaas (may Allaah be pleased with him) narrated that the Prophet (peace and blessings of Allaah be upon him) used to recite “Say (O Muslims): We believe in Allaah and that which has been sent down to us” [al-Baqarah 2:136] in the first rak’ah of Fajr and “Say (O Muhammad): O people of the Scripture (Jews and Christians): Come to a word that is just between us and you” [Aal ‘Imraan 3:64] in the second. But the Shaafa’is clearly stated that a complete soorah is better than a passage of equivalent length from a long soorah… This applies in prayers other than Taraweeh. But in Taraweeh reciting part of a lengthy soorah is better. They explained that by saying that the Sunnah in Taraweeh is to complete the whole Qur’aan in this prayer. End quote.
In conclusion: So long as your imam is not going to complete the Qur’aan in Taraweeh prayer, it is permissible for him to recite from various places in the Qur’aan and that is not makrooh, although it is better for him to recite a soorah in full. And Allaah knows best.