Uninstall Opera (don't choose to delete your data when uninstalling unless you want to). In both the admin account and the non-admin account, unpin any Opera icons from the taskbar and Start Menu that are left over. Delete any desktop shortcuts that are left over.
On the admin account, download the Opera installer, launch it, click "options", set "install path" to "C:\Program Files\Opera", set "install for" to "all users", adjust the other options as you want, and install.
Then things should work.
But, for a quick fix for the non-admin account, you'd just goto the install folder (should be "C:\Program Files\Opera"), right-click on launcher.exe and choose to create a shortcut on your desktop. Also, first, for the existing shortcut on your non-admin desktop, try deleting it to see if you can. If not because you need admin access, then leave that one alone. But, it'd be better to uninstall and reinstall to make sure things are right.