Shalmaneser III, during the 9th century BC, is very precise about the location of the city of Pitru: "At that time the town of Ashurutiraqbat, which the Hatti calls Pitru, which is on the Sagur River, on the other side of the Euphrates, and Mutkinu, who is on this side of the Euphrates - which Tiglath-pileser, the father, the prince, my predecessor, took by force, and which had been invaded at the time of Ashur-rabi II, king of Assyria, by the king of Aramu - these cities I restored and I installed Assyrian officials there."
Trevor Bryce notes that the Assyrian designation "Ana-Ashur-uter-asbat" presumably gave "Awsar", which is the name of a current municipality of Syria. For some researchers, it would be the city of Pethor in the Bible.
This city is not to be confused with that of Pitura in the country of Dirra, which lies rather to the north of the Tigris. On the other hand, it may be Pitura from the Hittite archives. In CTH41.1 Pitura is on the border between the Hatti and the Kizzuwatna on the water side (which can be understood as the sea or as the Euphrates). This is the first city mentioned as part of the Kizzuwatna.
My notes of veracity :
Pitru on the Sagur was the present town of Awsar (Tell Aushariya): 3/5