The Abrahamic religions have Biblical apocrypha, sections of the work claimed by some to be authentic but excluded by the main body of the religion. Does the Gita have any similar sections?
Hindu scriptures, including the Mahabharata which contains the Bhagavad Gita, were originally passed down through countless generations via oral tradition. But nowadays the main source we have on them (other than the limited oral tradition that still goes on in some ashrams) is via written manuscripts, and manuscripts had to be copied and recopied as the material they were written on (either bark in North India or palm leaves in South India) was of poor quality. So as it happens, the best manuscripts we have today of the Bhagavad Gita are not the manuscripts of the Mahabharata, but rather manuscripts of commentaries of the Bhagavad Gita, because a commentary needed to quote accurately the verses it was commenting on.
The famous Adi Shankaracharya wrote a commentary (the Bhagavad Gita Bhashya) which quoted 693 of the 700 verses of the Bhagavad Gita, so the "Shankara recension", the version of the Bhagavad Gita quoted in his commentary, as preserved in the South Indian manuscripts, has served as the accepted reference for verses in the Bhagavad Gita - it's the one relied upon whenever people make translations of the Bhagavad Gita or the Mahabarata (since the Mahabharata has to include the Bhagavad Gita).
But in parallel with the South Indian manuscripts of Adi Shankaracharya, there is another commentary on the Bhagavad Gita that has been diligently preserved in manuscripts - the commentary of the Kashmiri Shaivite philosopher Abhinavagupta. He commented on relatively few verses, but some of the verses he quotes are verses that we otherwise don't have from any other source. The version of the Bhagavad Gita quoted in Abhinavagupta's commentary, as passed down in Kashmiri manuscripts, is known as the "Kashmiri recension." In the introduction to this book, Shripad Krishna Belvalkar describes some of the differences between the Shankara and Kashmiri recensions. This mostly has to do with relatively minor grammatical and syntactical changes that lead Belvalkar to conclude that the Kashmiri recension is probably a distortion by later scribes of the original text of the Bhagavad Gita, which in his view would be closer to the Shankara recension.
But the more important difference is the fourteen extra verses unique to the Kashmiri recension. These would be the only thing that could possibly be called by the term "Bhagavad Gita apocrypha". That's really tiny, but you have to consider that the Bhagavad Gita is a 700-verse portion of a 100 thousand-verse epic poem, so there's not much room for apocrypha. In any case, if you want to know what these verses say, they're given in Sanskrit in the appendix of Belvalkar's book. But if you don't know Sanskrit, you can read Swami Lakshmanjoo's brief commentary on the extra verses. (Lakshmanjoo apparently believed that these few extra verses contained some important mystical truths that only the Kashmiri Shaivites were in possession of.)
As senshin mentions above, there is no fixed canon of Hindu scripture.
However, if you are referring to Hindu scripture whose authenticity is not held to be infallible, the smṛti parts of Hindu scripture might fit that description--in particular, the purāṇas, which are very sectarian scripture and have very obvious contradictions
I recommend that you research the difference between śruti and smṛti Hindu scripture, but keep in mind that not all Hindus (as with any point of Hindu faith!) make this distinction, or make this distinction as it is widely held.
The Bhagavadgītā itself technically falls into the śruti class of Hindu scripture, but this does not mean that it is not true or valid in any way. It simply signifies that the words are of human origin (but this does not mean the words are not divinely inspired).
But I think the best answer is that there is nothing like the Biblical Apocrypha in Hindu scripture, particularly in the Bhagavadgītā. There is only one Sanskrit version of the Bhagavadgītā. Only the translations, and sometimes the structuring of the lines, may vary.
"Apocrypha" is not an appropriate term when referring to the Gita.
Comparison with Bible
The Bible is a compendium of over 60 books. Bhagavad-gita is but a chapter in a canto of the epic Mahabharata.
In the biblical tradition many texts are anonymous and the origins and transmittance of the text obscure. Most Vedic texts mention the author (either at the beginning or at the end), even many songs include a verse about the author. When properly received, the texts come from an unbroken parampara (chain) of teachers-students where each member is a known (at least, to the adjacent generations) reliable saint. Mahabharata gives Vyasadeva as its author and he was still available in the 13th century to validate Madhvacarya's commentaries.
The very text of the Bible gets edited at times, but it is forbidden to modify standard shastra in the vedic tradition. Instead, sages write commentaries or even commentaries on commentaries, or their own treatises.
Abundant commentaries on Bhagavad-gita attest to its preservation. Shankaracarya's commentary is from 7th century, but his mission was to misinterpret the Vedas. Commentaries of the 4 Vaisnava sampradayas (who preserve the original message) are available here.
Substitutes for apocrypha
Somewhat similar in spirit, an "enhanced version" of Gita is Purnacandra's Gitamrita, based on the commentary of Vishvanatha Cakravarti Thakura.
There is another chapter in Mahabharata, where Arjuna asks Krishna to repeat the Gita again, but Krishna tells him some other story.
The 11th canto of Srimad-Bhagavatam contains Uddhava-gita, which is similar in topics to the Bhagavad-gita.
The very speaker of Gita -- Lord Krishna was present on the Earth from 1486 to 1534 A.D. in Bengal as Nimai pandita a.k.a. Krishna Caitanya. He was actively preaching his teachings again, so those could also be taken as an extension to the Bhagavad-gita. However, he did not indicate any discrepancies in the extant version of Gita.