As salamu aleykum,
Are there any narrations that prove that we should read ''bismillahirrahmanirrahim'' after Surah Fatiha and before the additional Surah which we read in Salaat ?
Please mention them.
Was salaam

All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His Slave and Messenger.

As far as we know, there is no Hadeeth proving that it is obligatory to read the Basmalah (Bismillaahi Ar-Rahmaani Ar-Raheem) before the Surah which follows Al-Faatihah in prayer. Muslim jurists held different opinions on the ruling of the Basmalah. Some scholars said that it is not at all a Sunnah act to read the Basmalah between Al-Faatihah and the following Surah. This is the opinion of Abu Haneefah and Abu Yoosuf . Other scholars made a distinction between the silent prayer and the prayer with loud recitation. They said that the Basmalah should be read in the silent prayer, not in the prayer with loud recitation, this is the opinion of Muhammad ibn Al-Hasan .

On the other hand, the Hanbali scholars are of the opinion that it is recommended to start the Surah which follows Al-Faatihah with the Basmalah, but silently because the Prophet, sallallaahu ‘alayhi wa sallam, used to say (Bismillaahi Ar-Rahmaani Ar-Raheem) silently in prayer.

Furthermore, some other scholars held different opinions on this issue. So, the matter should be taken at ease and whoever says the Basmalah before the Surah, then all the best, and whoever leaves it, there is hopefully no blame on him.