When I place a graphic into Illustrator (in this case from Photoshop, a black and white image of a glass), I trace it without issue, and then want to place it as the top layer on a black square.

First issue doing this, is that the placed item is in a frame that is not transparent, therefore there is a white box surrounding the glass, which stands out due to the black background.

Ok, to address this, used the advanced setting for trace to ignore white (windows>image trace: tick 'ignore white'). The graphic now has a transparent frame, but the white parts within the image itself are now also transparent.

What is the best way to resolve this? I was thinking of just making a copy of the now transparent object, fill this copy of the object so it is solid white, and put it as a layer beneath the fully transparent one.

This does not sound like the correct way to approach this, and besides, have been unable to create a solid white copy of the original in any case.

Any assistance greatly appreciated with the best practice for what I am trying to achieve, and also would like to know how to create solid white filled copy of the graphic. The graphic is enclosed within black outline with white on the inside and a black outline, so nothing overly complicated.