You can probably set the scanning area in your scanner and only scan the content that you need. If you want to scan the complete page and then remove the content you don't want, you have a few options:
If you have Adobe Acrobat Pro, you can use the Redaction tool to mark the area that you want to remove, set the redaction properties to fill the area with your page background color (e.g. white), and then apply the redaction marks. If you only have Adobe Acrobat Standard, and you don't care that the content you are trying to remove may still be in the file, but are happy with just covering it up, you can use the comment tools (e.g. the rectangle tool), set the fill and border color to white (or whatever your background color is) and cover up that content. This is not secure - somebody can easily (even after you flatten the page) uncover this hidden content).
You can also use an image editor (e.g. Photoshop), then use Tools>Content Editing>Edit Text&Images, right-click on your scanned image and select to edit the image in Photoshop. Now you can remove the content you don't want, save the image and it will automatically get imported back into Acrobat. If you now do a "Save As", the original content should be gone and replace with the modified content. I would only trust the redaction feature to securely remove content.
Karl Heinz Kremer
PDF Acrobatics Without a Net
PDF Software Development, Training and More...
http://www.khkonsulting.com