Unwanted border around png picture

What you could do is open the png file into photoshop, ctrl click or cmd click the thumbnail of the image.

Now you have a selection around your object and not the background. (assuming the png had a transparent background to start which it should have)

You now have a few options at your disposal.

1) You could create a mask from the selection then using a brush paint away the unwanted area in the mask. (black to hide, white to reveal (shades of gray to vary the amount)

2) You could use the menu item Select>Modify>Contract (use a low pixel number) or Select>Modify>Feather

3) You could use Select>Refine Edge - This will be more visual than #2

4) With 2 and 3 either make a mask or move to new layer (Another option is to invert the selection then hit delete) Mask is always preferable for noon-destructive edits that you can redo later assuming you save a master file as psd or tiff.

5)Instead of a selection you could try (on a layer that is unlocked) Layer>Matte>Defringe

After you save a master file, then you can merge the layers and save as a png file for the web. saving as a png would be merged anyways, however this will let you verify the end result. If not correct, you still have the master file to work from.