What needs to be done, if the caller to prayer makes a mistake while delivering the Adhan? The mistake was adding an extra Ish-had that there is no God but Allah. What Kaffara needs to be made, if any?

Praise be to Allah, the Lord of the Worlds; and may His blessings and peace be upon our Prophet Muhammad and upon all his Family and Companions.

Adding some Dhikr (remembrance of Allah) or some other words to the Adhan does not affect its validity. But the listener should not repeat the additions because they are not from Adhan.
Basing on this, the Adhan where its Muezzin added some Dhikr like Tashahud is correct and valid. And the person who performed it in that way did not commit a sin unless his intention was to innovate in Adhan adding something to it that is not proved by Sunnah. In that case, he commits a sin and should immediately stop doing that and repent to Allah.
Allah knows best.