It can be any type, historical, fantasy, comedy, melodrama

I am not sure  but maybe: 

Go Back Couple and Familiar Wife : In both, the MLs want to break up with the FLs, but they break up by traveling in time.

Cunning Single Lady :  In this one, the FL proposes to break up but the ML accepts too because they seem to be fed up with each other. They kinda have a love - hate relationship.

Emergency Couple : They break up annoyed with each other but fall in love again.

Here's what came to my mind:

  • The Love You Give Me  ML breaks up with FL over a misunderstanding, then meets her 5 years later.
  • True Beauty  ML breaks up with FL because he felt guilty over their extended separation while he was taking care of his father.
  • Love Like the Galaxy: Part 2 ML breaks up with FL after not wanting her involved in his complicated affairs.
  • Lighter & Princess ML breaks up with FL after seeking revenge for his sister's death.

How about FL reject ML?. 2023 My Dearest - Historical

A Korean drama where the male lead breaks up with the female lead and deeply regrets it is "Fated to Love You".  In this drama, the male lead, Lee Gun, initially enters a contract marriage with Kim Mi-young, but his pride and past trauma lead him to break things off with her. He later deeply regrets his decision and spends the rest of the series trying to win her back, facing numerous obstacles.