Basically these scenarios :

-Naive/Cute guy and a Protective girl

-Cheeky/Childish guy and a Badass girl.

-Cute, innocent guy and Cold or caring girl

I know there might not be many or any recs at all since I haven't come across one. But my introverted boss was ok. Plz gimme some fluff like this. I'm waiting

I don't know if these are exactly what you're looking for, but they all have the FL protecting the ML as a strong element in the drama. 

Shopping King Louie - both of the leads were naive but the FL way less than the ML. She took care of him basically the whole drama. It was adorable. 

Strong Woman Bong Soon - strong cute girl protects rich weak guy. 

Oh My General - badass tomboy protects weak and childish husband

The Legends - The FL only protects the ML at the beginning. It's a shame cuz she's so badass initially, but they end up throwing her into the weak FL being protected by the ML for most of the drama. But man, the FL was super rad and the ML was an adorable puppy for a while there at the beginning. 

Hotel del Luna - I'm not done with it but the cold FL has saved the warm ML's ass a few times already.

Part of Empress Ki (bodyguard at the beginning, 2nd lead, kinda... depends who you ship)

He is is Psychometric (Just the cold/cute dynamic, but she is a police officer)


Maybe: Protect the boss - (haven't watched completely but she is literally his bodyguard)

                   Sweet Combat (Kickass and has a cold female Mr Darcy thing going on)

                  An Oriental Odyssey (Certainly saves him in the beginning and is his master)

Dk if this is what you meant... but hey, gave it a try! Hope you find something you like :)