These are only partial matches I think, I think you're looking for sort of funny cinderella story where the FL takes pity on the ML first with a charade ensuing.
Shopping King Louie (rich guy with amnesia, rescued by a poor girl that thinks he is homeless)
Terius Behind Me (he is a spy, but she thinks he's a jobless guy that becomes a nanny for her kids)
The King: Eternal Monarch (she is a detective, arrests him thinking he is some crazy person or homeless)
Dong Yi (she thinks he's just a police officer, but he's the king and she is a slave in the palace)
100 Days My Prince (another amnesia story, this time the guy is the crown prince who doesn't know who he is, forced into a marriage with a strange woman...)
I'd like some recommendations on this kinda drama as well, I thought I had watched a lot of these, but clearly I have not!