How can the screen black border smudge be fixed?

There is no fix on this without replacing the screen. I know exactly what you are talking about.

This occurs on iPhone 5 series screens where you see a yellow border around the screen on a white background. For some reason the LOCA liquid transparent glue that goes between the screen layer and the glass causes discolouration on the screen over time.

The only way to fix this is to replace the screen. And IMO the screen is already a cheap fix, you can easily buy a replacement screen online for as little as $30, you can definitely get higher quality ones for $50.

iFixit does sell replacement screens for the iPhone 5s including all the tools you need to open and replace the part together.

Where to get the replacement part and tools required: iPhone 5s Screen

Guide to replace part:

iPhone 5s Screen Replacement