Hi!
To hide the settings icon, go to the Frontend search settings -> General panel: https://i.imgur.com/4DWssHw.png
There is no option for opening the settings on input focus, but with a simple javascript snippet using the plugin API it’s possible. I’ve quickly put together a possible solution for you. Just include it in the functions.php file in your theme (or preferably child theme) directory (copy from line 3):
There is also an addition to hide the settings once the results are displayed. If you want to learn more about the search JS api, you can find it here.
Let’s hope this works!
Best,Ernest Marcinko
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