At a Distance Spring Is Green  (ML is not weak he just had a past trauma so FL is  very warmth and supportive)

Cinderella and The Four Knights (You have one girl supporting the other MLs, they don't rely on her at first though)

Rookie Historian Goo Hae Ryung (due to social condition ML is  not in contact with much people so he's drawn by FL behaviour)

The thing they have in common is the FL is really the light IMO :P

There's also Meeting You - total fluff but I think it fits. The ML is not cold, just very awkward around everyone except to FL.

 lanzhanya:

Any dramas where the female lead is the only light/support the male lead has and is heavily reliant on her warmth? I don't mean a weak male lead but maybe a male lead that is misunderstood by others or is cold and no one dares to come close to him? Like TTJ from Till The End of The Moon. 

lighter and princess