Original png on left. How can i avoid this?
Alpha values range from 0-255. Your image editing program is hiding pixels with very low values such as 1-3 into that background checkerboard pattern. Blender is expecting 0 for complete transparency.
Open image in GIMP, use the magic wand tool to select transparent area, then click Delete key.
File >> Export As... and give it a new name, or Overwrite PNG if you are satisfied with results.