Muhammad reported on the authority of some of his teachers that Ubayy b. Ka'b led them in prayer during Ramadan. He used to recite the supplication (in the witr) during the second half of Ramadan. (Using translation from Abū Dāʾūd 1428)   

أَنَّ أُبَيَّ بْنَ كَعْبٍ أَمَّهُمْ يَعْنِي فِي رَمَضَانَ وَكَانَ يَقْنُتُ فِي النِّصْفِ الْأَخِيرِ مِنْ رَمَضَانَ  

[Machine] Ibn Sirin disliked performing the Qunut (supplication) during Witr prayer except during the last half of Ramadan.  

كَانَ ابْنُ سِيرِينَ يَكْرَهُ الْقُنُوتَ فِي الْوِتْرِ إِلَّا فِي النِّصْفِ الْأَوَاخِرِ مِنْ رَمَضَانَ  

الْقُنُوتُ فِي النِّصْفِ الْأَوَاخِرِ مِنْ رَمَضَانَ  

"عن الحارثِ، عن علىٍّ: أنهُ كانَ يقنُتُ فِى النِّصْفِ الأخيرِ منْ رمضانَ".