If you are referring to the search suggestions that appear below the general search field on most eBay pages when you begin entering search keywords, you can delete the suggestions from previous searches by going to your Recently Viewed items page and using the "Clear all" link there:
https://www.ebay.com/mye/myebay/rvi?key=recentlyviewed
Once you have done that, the only suggestions you should see when searching are generic suggestions that have nothing to do with your previous searches.
The Advanced Search page has a similar search field that operates differently; the suggestions shown there are saved form data from your browser. You can delete them all at once by going to your browser settings are erasing form data there, or individually by using the arrow keys to select a suggested search, and using delete or shift-delete to remove the suggestion.
If you are referring to some other search suggestions, please describe further what you are trying to remove.
