Oliver Cowdery, the scribe here in 𝓟, initially wrote “the king’s army” (although the possessive form Kings was written without the apostrophe). Oliver immediately caught his error here, crossed out Kings, and then continued inline with “of the king”. The original manuscript undoubtedly read “the army of the king”. Of course, either genitive form is possible. Oliver’s initial error was probably the result of the preceding instance of “the king’s army” that occurs only a couple of verses earlier:
The text has two instances of “the armies of king Noah” (both in Mosiah 23:1), but no more of “the king’s army” (or “the king’s armies”). In Mosiah 18:34, “the king’s army” avoids the rather awkward “apprised of the coming of the army of the king” (that is, a sequence of three prepositional phrases beginning with of ).
Summary: Accept in Mosiah 19:1 Oliver Cowdery’s immediately corrected reading in 𝓟, “the army of the king” rather than “the king’s army”, which occurs two verses earlier in the text.