The 1837 edition dropped the word again. Joseph Smith did not mark this change in the printer’s manuscript. This change appears to be a simple typo, especially since this is not the first time that Abinadi has prophesied to the people of Noah:
Thus there is nothing inappropriate about referring to Abinadi beginning to prophesy a second time. Moreover, there are 14 other examples of “began again” in the Book of Mormon text, including one more that refers to speaking: “therefore he began again to speak unto them” (Helaman 8:10).
Summary: Restore the again in Mosiah 12:1, the earliest extant reading (“Abinadi … began again to prophesy among them”).