Tsuki ga Kirei Special is just a compilation of skits that focuses on the supporting characters, most of which are the secondary couples. The skits don't add anything to the plot so anyone can watch it while also watching the main series; I watched this after I finished the main episodes just to seal off my journey watching this series.
The skits are pretty hilarious, especially the ones that focus on Kotarou's parents. If there is something I hope the special could have focused more on, it would be Ryokou-sensei and Roman's relationship because the main series already introduced them as what we already know but failed to capitalize it.
There is nothing else I can say about Tsuki ga Kirei Special other than it's short and supplementary though I'd definitely recommend it to anyone who just picked up the series. A little love for the supporting characters is not too much to ask, right?