The widget for FileField and ImageField has a problem: it doesn't supports clear its value and it doesn't delete the old file when you replace it for a new one.
This is a solution for this. It is just for Admin, but you can make changes to be compatible with common forms.
The jQuery code will put an <input type="checkbox"> tag next to every <input type="file"> and user can check it to clear the field value.
When a user just replace the current file for a new one, the old file will be deleted.