Remove the X icon inside the search box of Search Module

We want to remove the X icon that appears on the search box when you type a search keyword. We checked the generated HTML and can't seem to find it there.

Is there a way where we can select it via CSS to make it display none?
Or a way in JS code to disable it from appearing when you type in a search keyword?

I have attached a screenshot of what I'm talking about.
Help is really much appreciated.