If you want to make the image have a transparent background, erase the background to be completely white. Then Select the Magic Wand Tool, set the Tolerance to 0, and untick Anti-Aliasing and Contagious. Click on the background and hit the DELETE key. ...
I have another graphic with a logo that Express butchered (I overwrote the original exported file with the following fix, but it's hardly a fix, and I'm not reverting just to export an example). The logo was imported as atransparent background PNG. It looks retched in Express, e...