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.