Hi stars-are-pink and welcome.
Right-click on a blank part of your Desktop and select "New Shortcut".
In the 'Location' dialogue box, type in the URL for Hotmail - "www.hotmail.co.uk" (or "www.hotmail.com" if your account is there) and click 'Next'.
Enter a name for the shortcut like 'Hotmail' and click 'Finish'
This should put a 'Hotmail' Icon on your DesKtop - to put it into Vaio Gate, just click on the shortcut and drag-and-drop it into the Vaio Gate area of the screen.
Clicking on this new shortcut should open Hotmail in your default browser.
