Where is the reference to John's student writing that John wrote his gospel because the synoptic gospels needed more detail? Two early church fathers (Ignatius and Polycarp) were students of John the apostle.  Someone wrote that one of them wrote that John was unhappy with missing details of the previous gospels, motivating him to write his gospel.  I've been trying to find this reference in their writings.
Not aware of the idea that John wrote his gospel because the other three needed added details, but there is the thought that John wrote for more &quot;light&quot; on certain general matters.  These are from Irenaeus, a disciple of Polycarp, a disciple of John.


In a still clearer light has John, in the Apocalypse, indicated to the Lord’s disciples what shall happen in the last times, and concerning the ten kings who shall then arise, among whom the empire which now rules [the earth] shall be partitioned.
Irenaeus AH V XXVI


Irenaeus in AH III XI does argue that the gospels must be 4 in number and does outline why John's gospel is so important.


John, the disciple of the Lord, preaches this faith, and seeks, by the proclamation of the Gospel, to remove that error which by Cerinthus had been disseminated among men, and a long time previously by those termed Nicolaitans, who are an offset of that “knowledge” falsely so called, that he might confound them, and persuade them that there is but one God,
-AH III XI-


As the OP mentions, John taught Polycarp and Polycarp taught Irenaeus.  Here is an example.
TO EDIT TO INCLUDE EUSEBIUS' EXPLANATION (mentioned in another answer)


And when Mark and Luke had already published their Gospels,762 they say that John, who had employed all his time in proclaiming the Gospel orally, finally proceeded to write for the following reason. The three Gospels already mentioned having come into the hands of all and into his own too, they say that he accepted them and bore witness to their truthfulness; but that there was lacking in them an account of the deeds done by Christ at the beginning of his ministry.763




And this indeed is true. For it is evident that the three evangelists recorded only the deeds done by the Saviour for one year after the imprisonment of John the Baptist,764 and indicated this in the beginning of their account.




For Matthew, after the forty days’ fast and the temptation which followed it, indicates the chronology of his work when he says: “Now when he heard that John was delivered up he withdrew from Judea into Galilee.”765




Mark likewise says: “Now after that John was delivered up Jesus came into Galilee.”766 And Luke, before commencing his account of the deeds of Jesus, similarly marks the time, when he says that Herod, “adding to all the evil deeds which he had done, shut up John in prison.”767




They say, therefore, that the apostle John, being asked to do it for this reason, gave in his Gospel an account of the period which had been omitted by the earlier evangelists, and of the deeds done by the Saviour during that period; that is, of those which were done before the imprisonment of the Baptist. And this is indicated by him, they say, in the following words: “This beginning of miracles did Jesus”;768 and again when he refers to the Baptist, in the midst of the deeds of Jesus, as still baptizing in Ænon near Salim;769 where he states the matter clearly in the words: “For John was not yet cast into prison.”770




John accordingly, in his Gospel, records the deeds of Christ which were performed before the Baptist was cast into prison, but the other three evangelists mention the events which happened after that time.
-source-