Neo-Platonism (http://en.wikipedia.org/wiki/Neoplatonism_and_Christianity). Augustine was influenced by Neo-Platonism, as well as other Christian writers. I think you will find Neo-Platonism to be more more influential than Platonism.
I am not expert in Philosophy enough to provide my own explanation. All I can do is provide some references. From the Stanford Encyclopedia of Philosophy (http://plato.stanford.edu/), Augustine is described as a Christian Neoplatonist. Augustine is highly influential in later times as well and, as stated in the below, also being a main figure in the merging of Greek philosophical tradition and the Judeo-Christian religious and scriptural traditions. So, this would show influence of Neoplatonism on the early Church.
From the Stanford Enclcyclopedia of Philosophy (http://plato.stanford.edu/entries/augustine/) (note that I have added boldface type): "Aurelius Augustinus [more commonly “St. Augustine of Hippo,” often simply “Augustine”] (354–430 C.E.): rhetor, Christian Neoplatonist, North African Bishop, Doctor of the Roman Catholic Church. One of the decisive developments in the western philosophical tradition was the eventually widespread merging of the Greek philosophical tradition and the Judeo-Christian religious and scriptural traditions. Augustine is one of the main figures through and by whom this merging was accomplished. He is, as well, one of the towering figures of medieval philosophy whose authority and thought came to exert a pervasive and enduring influence well into the modern period (e.g. Descartes and especially Malebranche), and even up to the present day, especially among those sympathetic to the religious tradition which he helped to shape (e.g. Plantinga 1992; Adams 1999).
Plotinus is generally regarded as the founder of Neoplatonism. For more about him see http://plato.stanford.edu/entries/plotinus/
From this article you can begin to get an idea of what Neoplatonism is: "The term ‘Neoplatonism’ is an invention of early 19th century European scholarship and indicates the penchant of historians for dividing ‘periods’ in history. In this case, the term was intended to indicate that Plotinus initiated a new phase in the development of the Platonic tradition. What this ‘newness’ amounted to, if anything, is controversial, largely because one’s assessment of it depends upon one's assessment of what Platonism is. In fact, Plotinus (like all his successors) regarded himself simply as a Platonist, that is, as an expositor and defender of the philosophical position whose greatest exponent was Plato himself. Originality was thus not held as a premium by Plotinus. Nevertheless, Plotinus realized that Plato needed to be interpreted. In addition, between Plato and himself, Plotinus found roughly 600 years of philosophical writing, much of it reflecting engagement with Plato and the tradition of philosophy he initiated. Consequently, there were at least two avenues for originality open to Plotinus, even if it was not his intention to say fundamentally new things"
Also from the same article: "The three basic principles of Plotinus' metaphysics are called by him ‘the One’ (or, equivalently, ‘the Good’), Intellect, and Soul (see V 1; V 9.). These principles are both ultimate ontological realities and explanatory principles. Plotinus believed that they were recognized by Plato as such, as well as by the entire subsequent Platonic tradition."
More about Plotinus here (Internet Encyclopedia of Philosophy) http://www.iep.utm.edu/neoplato/
This except shows Plotinus to be a synthesizer of various philosophical schools.
"In the last analysis, what stands as the most important and impressive accomplishment of Plotinus is the manner in which he synthesized the pure, ‘semi-mythical’ expression of Plato with the logical rigors of the Peripatetic and Stoic schools, yet without losing sight of philosophy’s most important task: of rendering the human experience in intelligible and analyzable terms. That Plotinus’ thought had to take the ‘detour’ through such wildly mystical and speculative paths as Gnosticism and Christian salvation theology is only proof of his clear-sightedness, thoroughness, and admirable humanism. For all of his dialectical difficulties and perambulations, Plotinus’ sole concern is with the well-being (eudaimonia) of the human soul. This is, of course, to be understood as an intellectual, as opposed to a merely physical or even emotional well-being, for Plotinus was not concerned with the temporary or the temporal. The striving of the human mind for a mode of existence more suited to its intuited potential than the ephemeral possibilities of this material realm, while admittedly a striving born of temporality, is nonetheless directed toward atemporal and divine perfection."
Another description of Neoplatonism can be found at New Advent (http://www.newadvent.org/cathen/10742b.htm).
"A system of idealistic, spiritualistic philosophy, tending towards mysticism, which flourished in the pagan world of Greece and Rome during the first centuries of the Christian era. It is of interest and importance, not merely because it is the last attempt of Greek thought to rehabilitate itself and restore its exhausted vitality by recourse to Oriental religious ideas, but also because it definitely entered the service of pagan polytheism and was used as a weapon against Christianity. It derives its name from the fact that its first representatives drew their inspiration from Plato's doctrines, although it is well known that many of the treatises on which they relied are not genuine works of Plato. It originated in Egypt, a circumstance which would, of itself, indicate that while the system was a characteristic product of the Hellenistic spirit, it was largely influenced by the religious ideals and mystic tendencies of Oriental thought."
Hopefully, the above will provide and idea that Neo-Platonism is not Platonism and that it had influence on early Christian writers most notably Augustine.