Material Function for blending colours in a texture? 4-25

Each frame I am drawing to a render target which is passed to a material. Here’s an example:
Texture-Before

It’s a pixel game so the low res is desired, but I want to soften the black area along the edge of the white area. I want it to look something like this:
Texture-after

Anyone know a way to achieve something like this?

Hey guys, I’m still struggling with this one. I’ve been looking everywhere for something to help but haven’t found anything. Any suggestions are appreciated.