Wow, this thread has some trash talk. I'm not going to argue with other contractors on this site, I have my own ways and I backup my work. And on top of that I'd be more than willing to provide pictures before, during and after regarding repairs like this.
Something like what has been posted is extremely common amongst flat roofs. The metal flashing that you see on the perimeter of the low slope area is called "cap flashing" or "perimeter flashing" depending on the contractor you're speaking to. While in some cases minor damage can be covered up with caulking and painted over, I recommend full replacement of the fascia. I'm not trying to promote myself, my business or my practise but I would only give the customer the option of replacing the entire length of fascia that has been damaged and replacing the cap flashing as well.
Opt for custom made cap flashing that has a "kickout" or a hemmed lip that allows water to drip away from the fascia. I recommend a minimum thickness of 26-gauge flashing with a height that will cover the entire fascia. Therefore you will never need to repaint your fascia every couple of years. I have this on my own home, although not particularly a flat roof it serves the same purpose.