The origin of this problem is the transparent background or the rollover element. If you would have made its background color the same as the icon background color, there would have been no overlapping.
To correct the problem please try adding the following rules to the end of your custom CSS theme (Ecwid control panel -> ‘System settings’ tab -> Design -> CSS themes):
border: 0 none;
Click "Save (Ctrl+S)".
The first rule sets the white background, that lest you place the rollover image above the inactive state of the cart image. It also sets the height, that fits the image size, and removes the border of the element displayed on rollover.
The second rule sets the element (Minicart-1) , the position of the minicart icon should be defined according to.
The third rule sets the absolute position of the minicart link.