Relatively Weak Male MC with a Relatively Strong Female Lead

Strong Woman Do Bong Soon (physically strong female lead)

Witch's Court (really smart and sassy female lead)

Bring It On, Ghost (physically strong female lead)

Reply 1988 (confident female lead, shy male lead <-- depends on who you view as the male lead though)

Shopping King Louie

I haven't watched this myself yet, but this was recommended by Dramabeans in this forced cohabitation article (a trope I love):  

That Fool (aka Accidental Couple):  Accidental Couple (also known as That Fool), in which a movie star hires a regular guy to pose as her husband in order to avoid a scandal with her actual boyfriend. The hero clearly recognizes that it’s a business deal and his “wife” is out of his league, yet he still demands respect when she tries to dismiss him the way she’s used to doing with everyone. His quiet dignity is swoonworthy (props to Hwang Jung-min for an excellent performance), and as viewers we fall in love with his decency along with the heroine, as she slowly begins to realize what a treasure she has by her side, and decides to fight for him.

• Drama: https://kisskh.at/371-that-fool
• The full article:  http://www.dramabeans.com/2017/06/dramaland-catnip-marriage-before-dating/

• Ms. Temper & Nam Jung Gi
• That Fool
• Boku to Star no 99 Nichi
• Densha Otoko
• Tatta Hitotsu no Koi
• The Innocent Man

Across the Ocean to See You - female lead is male lead's superior in their company

Date - female lead is technically much more accomplished than male lead(?)

 Rin:

Across the Ocean to See You - female lead is male lead's superior in their company

Date - female lead is technically much more accomplished than male lead(?)

is there more of show like Across the ocean to see you?