If you are not in 16 million colours, then when you cut, the background will fill in automatically. If you are in 16 million colours, then if you have grey around the edges, it's the colour that your PSP is assigning to "transparent". Make a new layer and fill it with some horrible colour and then drag it to under the part you want to see, to check if it is really transparent.

If you are not in 16 million colours, go to color - increase colours - 16 million.

If you want to make a background transparent for a gif, then fill in any "transparent" spots with a specific colour. Then make that same colour the background colour. (be careful to use a colour that is not used anywhere else in the image, or you will have interesting holes...) Then go to color - set palette transparency.

It will ask if you wish to convert the picture to a single layer and 256 colours. Say yes, then experiment with which method of reducing colour works best. I usually use ocree. Then, "set transparency to background color". Then save as a GIF and not as psp or jpg. This is usually the last stage I do in editing, and I save in psp format often.

If it doesn't work quite the way you want, well, that's what the undo is for.

Have fun!