Last night I noticed a couple of admirality assignments for my main starfleet toon not awarding XP for progressing that campaign, but I assumed that it was just that I was forgetful and selected one that was pure specialization xp. Today, I have had all of my Klingon Empire assignments for that toon (haven't checked others yet) not awarding xp for the campaign. I am currently stuck at 65,288/67,900 xp for reaching level 10 with it.
As far as I can tell, the others work fine because the Romulan assignments still award xp.
Edit: Apparently other characters can still earn Klingon admirality xp for me, just not the one that is almost level 10 with that campaign.