Is there a way to add an icon to a pinned Internet Explorer page on the task-bar? One of the sites I have pinned does not and it is very ugly and I have no idea what site it is because the icon is just a white square.
Yes, you can modify the shortcuts. Simply, they are icons which you can modify their properties to change to an icon you prefer.
Here are more detailed instructions if you aren't familiar with how to do this; How to Customize Your Windows 7 Taskbar Icons for Any App.