Praise be to Allah.
1 – Reciting Qur’aan after al-Faatihah in prayer is not obligatory, either in a fard (obligatory) prayer or a naafil (supererogatory) prayer, whether the prayer is one in which Qur’aan is recited out loud or one in which it is recited silently, whether one is joining the prayer late or otherwise.
It was narrated that ‘Ata’ said: Abu Hurayrah said: In every prayer there is recitation. What we heard the Prophet (peace and blessings of Allaah be upon him) say, we tell you, and what he did not tell us about we cannot tell you. Whoever recites Umm al-Kitaab (i.e., Soorat al-Faatihah) has done enough, and whoever does more than that, that is better.
Narrated by al-Bukhaari, 738; Muslim, 396.
The phrase, “Whoever recites Umm al-Kitaab (i.e., Soorat al-Faatihah) has done enough, and whoever does more than that, that is better” indicates that reciting al-Faatihah is obligatory, and that nothing else will do.
It also indicates that reciting a soorah after al-Faatihah is mustahabb. There is scholarly consensus on this point with regard to Fajr, Jumu’ah and the first two rak’ahs of every prayer, and this is Sunnah according to all scholars. Al-Qaadi ‘Iyaad (may Allaah have mercy on him) narrated that some of the companions of Maalik said that reciting a second soorah is obligatory, but this is an odd view which is to be rejected.
With regard to reciting an additional soorah in the third and fourth rak’ahs, there was a difference of opinion among the scholars as to whether it is mustahabb or not. Maalik (may Allaah have mercy on him) regarded it as makrooh. Al-Shaafa’i (may Allaah be pleased with him) regarded it as mustahabb in his new opinion but not in his old opinion; the old opinion in this case is more correct.
Sharh Muslim, 4/105, 106.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said:
According to the majority of scholars, reciting another soorah (after al-Faatihah) is Sunnah and is not obligatory, because it is obligatory to recite only al-Faatihah.
Al-Sharh al-Mumti’, 3/103
If you stand up to complete your prayer after the imam has said the tasleem, then the correct scholarly view is that the rak’ahs in which you caught up with the imam form the first part of your prayer. See question no. 23426. If what remains of your prayer is the third and fourth rak’ahs, then you should recite al-Faatihah only; if what remains is the second and subsequent rak’ahs, then in the second you should recite al-Faatihah plus another soorah, and in the remaining rak’ahs you should recite al-Faatihah only.
It is permissible for a worshipper to recite another soorah after al-Faatihah in the third and fourth rak’ahs, but he should do that only sometimes; it was narrated that the Prophet (peace and blessings of Allaah be upon him) did this according to a hadeeth narrated by Muslim (452).
And Allaah knows best.