I had my roof replaced about 5 years ago. I recently noticed that the shingles overhang most of the roof by 3/4 inch and the water runs directly off.
But in the middle part of the fascia on the back of my home the shingles are flush with the drip edge.
The drip edge lays flat against the top of the fascia, this allows water to actually run down the fascia getting it wet, And some parts are actually rotten.

What can i put there to make the water run off the roof like all the rest of the house?

Here is a link to a youtube video i took of the edge for you.

https://www.youtube.com/watch?v=DuIEwj5jIPk&list=UUZHJJ05ZqBHRLkpCwF1-d3A