Which school of philosophy influenced St Augustine primarily?
This is a great question. There were many schools of thought that influence the great St. Augustine.
First, we can say that Augustine was thoroughly steeped in classical education. For this reason, he was influenced by the great classical authors and traditions of his day and age. He was, therefore, influenced by Stoicism, which was popular among the wealthy. In addition, he was influenced by the teachings of the great philosopher, Plato. In fact, during this time, there was sort of renaissance in the thoughts of Plato.
Second, we can say that other classical authors had a profound influence on him. We can say authors like Cicero, Virgil, and other great Roman authors impacted his thinking. For example, we know what Varro said mostly through Augustine, as Varro is only extant in Augustine's text.
Finally, from a more religious point of view, he was influenced by the teaching of Mani. For a time, he was a devotee of Manichaeism.