How is it possible to delete a Google search or a search made via the
Search icon shown above the web address?  I would like some privacy
and do not want the name of my search to appear in the vertical column
that decends from the Google arrow.  Thanks

Hi there,

It sounds like you are refering to the Google Toolbar. If this is not
the case, just ask for a clarification.

At the very left of the Google Toolbar, left-click on the
multi-colored Google logo. In the menu that appears, click on Toolbar
Options...

Uncheck the box that says "Keep a search history in a drop-down list
under the search box".

Click on OK at the bottom of the page.

You should now have privacy.

Best wishes,
robertskelton-ga

I hope this works in practice.  I am pretty (very) new to all this. 
Thanks for getting back right away.