Scribe 3 of 𝓞 does not tend to add superfluous s ’s. Each king has his own reign, so the plural is perfectly acceptable here. Nonetheless, this usage was strange for Oliver Cowdery, so for both occurrences of reigns in 𝓞 (miswritten once as reings), he changed the plural to the singular reign when he copied this passage from 𝓞 into 𝓟. This same original plural usage occurs elsewhere in the small plates (four times), and in these cases the plural forms have been retained:
the reigns of the kings Jacob 1:9, Jacob 1:11, Jacob 1:14
the reigns of their kings Jacob 3:13
With one exception, the singular reign occurs only when the text describes the rule of a single king (21 times). The exception is found at the very end of the book of Mosiah, where Mormon is describing the replacement of the hereditary system of kings with elected judges:
This expression parallels the common Book of Mormon expression “the reign of the judges” (which occurs 102 times in the original text, from Mosiah 29 through Helaman 16). The plural use of reigns is, in fact, never found when referring to the judges. The first two examples of “the reign of the judges” are near Mosiah 29:47:
It is possible that the singular reign in Mosiah 29:47 (namely, “the reign of the kings”) is a mistake due to the nearby influence of these two examples of “the reign of the judges”. But a more plausible explanation is that the word reign in Mosiah 29, unlike its usage in the small plates of Nephi, specifically refers to the system of rule (either under hereditary monarchs or under elected judges) rather than the particular rule of a king or judge. Under this interpretation, the singular reign is wholly appropriate for kings as well as judges. Thus there is no strong motivation to emend reign in Mosiah 29:47 to read in the plural.
Summary: Restore the plural reigns both times in 1 Nephi 9:4; the small plates of Nephi always refer to “the reigns of the(ir) kings”; in Mosiah 29:47, the singular reign in “the reign of the kings” should be retained because it parallels “the reign of the judges” found throughout much of Mormon’s abridgment of the large plates of Nephi; in these later expressions, the word reign refers to the system of governance, never to the specific rule of a particular king or chief judge.