It's a heartwarming youkai show, in which our hero and his associate go around helping youkai. Its episodes are kind and compassionate. The main character's a thoroughly nice chap whose main flaw is a tendency to overempathise. (That's in his colleague's opinion, anyway.) It's rather lovely on a regular basis. However there are plenty of shows like that, e.g. Natsume's Book of Friends, so this show's point of distinction is that its title character is grumpy.
Hanae Ashiya is an unremarkable schoolboy who's kind-hearted, easily scared and a bit of a crybaby.
Haruitsuki Abeno works as a youkai exorcist, although this is always done with the youkai's interests at heart and with its permission. "Exorcism" means "helping it return to the youkai dimension". It's his job, although I don't think we ever see him accepting or demanding payment from anyone except Ashiya. He's as human as you or me, but he doesn't seem to like people and he's got a bit of a reputation. He talks to Ashiya as if ordering around a slave. He's obnoxious, frankly, but he means well underneath and he cares a lot about youkai. He also has some notorious backstory that we never learn and might, perhaps, get addressed if there's ever a Season 2. (Alternatively, I could read the manga.)
He's the morose person of the title, but confusingly he's not the Mononokean. "Mononoke" means ghost or spectre (as in Hayao Miyazaki's Princess Mononoke) and this particular kanji for "an" means a hermitage or retreat. It's a safe space for youkai, basically. The Mononokean is a teleporting room. She can materialise her door in any wall that's big enough, letting you travel anywhere at all (including other dimensions). She's technically another youkai and she can talk to you by making words appear on a hanging scroll, despite having no body beyond the obvious one of four walls, a ceiling and a floor. Had this been SF, she'd have been a disembodied AI. (I only call her female because her wall scroll "talks" in feminine language.)
I can't imagine anyone calling this show unmissable. It doesn't really have a story. The small amount of story it does have (Abeno's past) is never revealed to us. The season finale does the obvious thing for this genre with Ashiya, then of course resolves it in the inevitable way. It's basically pleasant, except that the show's undercutting this with occasional bits of sandpaper. Most of this comes from Abeno. Some of the supporting characters are also capable of being abrasive (e.g. Zenko's father, or the childish fox youkai), while it's also demonstrated that youkai can be dangerous to your health, even if they didn't mean to be.
There's not a lot to say about this one. It's nice. I think the sandpaper is prominent enough that you're relatively unlikely to fall in love with the show, but you can confidently expect its episodes to end in a warm, compassionate way. The abrasive characters will always be nice underneath, really, while there are plenty of other people who are just nice through and through (Ashiya, Fuzzball, Zenko, almost everyone in the Underworld). The show's fine. It's a comforting thing to fill 25 minutes. You probably won't feel the need to rush off and watch all the episodes in order, but you'll end up enjoying any random episode you might happen to put on.