Well 3 and 4 Esdras and the Prayer of Manasseh were in the Vulgate for centuries. They weren’t part of the Canon of Trent but nonetheless Pope Clement Vlll put them in an appendix of the Clementine Vulgate in 1592 where they always were printed and still are in reproductions. The Nova Vulgate doesn’t include them.
But this explains why the original KJV includes these three texts with the deuterocanonical books as the Apocrypha and is part of that list even to this day( though they call 3 and 4 Esdras confusingly 1 and 2 Esdras). Actually the Anglican Communion considers 1(3) Esdras,2(4) Esdras, and the Prayer of Manasseh the same authority as any of the deuterocanonical books.
Source- Anglican 39 Articles of Religion.
Of the Names and Number of the Canonical Books.
Genesis, The First Book of Samuel, The Book of Esther,Exodus, The Second Book of Samuel, The Book of Job,Leviticus, The First Book of Kings, The Psalms,Numbers, The Second Book of Kings, The Proverbs,Deuteronomy, The First Book of Chronicles, Ecclesiastes or Preacher,Joshua, The Second Book of Chronicles, Cantica, or Songs of Solomon,Judges, The First Book of Esdras, Four Prophets the greater,Ruth, The Second Book of Esdras, Twelve Prophets the less.
And the other Books (as Hierome saith) the Church doth read for example of life and instruction of manners; but yet doth it not apply them to establish any doctrine; such are these following:
The Third Book of Esdras, The rest of the Book of Esther,The Fourth Book of Esdras, The Book of Wisdom,The Book of Tobias, Jesus the Son of Sirach,The Book of Judith, Baruch the Prophet,
The Song of the Three Children, The Prayer of Manasses,The Story of Susanna, The First Book of Maccabees,Of Bel and the Dragon, The Second Book of Maccabees.
All the Books of the New Testament, as they are commonly received, we do receive, and account them Canonical.