Not really. There was no "story arc," and there was never an explicit statement of a "second 5-year mission." There were just a handful of novels that claimed to be several years after TOS while still being pre-TMP, and the "second 5YM" theory is something fans made up retroactively to rationalize it. I discussed the idea in a thread in this forum about the '80s continuity some years back, as an attempt to postulate a unifying theory for books that were far less unified in actuality, and I fear many people have mistaken my after-the-fact analysis for something that was present in the actual books. If so, I regret being responsible for contributing to a false belief.
Black Fire certainly never said anything about a second 5-year mission. It did postulate that the bridge was rebuilt and the uniforms redesigned, presumably to suggest a transitional stage between TOS and TMP, but its descriptions of the uniforms and sets were at most only vaguely reminiscent of TMP. (I recall something about the new uniforms being mostly white, which was only true of Kirk's admiral uniform in TMP.)