The ‘white’ lines at edges of texture with transparency is a common issue for transparent image (not restricted to UE4). UE4 and many program apply Alising to texture, so often, the transparent (Alpha) & the image do not match 100%. result in what you have.
The simpliest way is to over color the border of the transpareny zone (in the background), Means if your border is dark brown, paint a little bit over the transparent zone on the under layer with brown. This way, even when the alpha & image border do not match perfectly, you still get the correct border colors.
Photoshop have a free plugin that that automate this for you, I look up when I get home.
Edit: Here is its. Check the video (Its mention Unity, but its more or less the same with UE4):
The plugin itself can be found here.
http://www.flamingpear.com/ Go to download & pick free plugin, solidify can be found inside. You can also check out the other plugin if you fancy.