There is no way to clear the search text from the native search bar, so in order to get what you want you would have to implement a custom search/filter.
This is fairly easy to do, the general approach is as follows:
Start with a details screen, and add your card list as an inline list component
Create a user specific text column in your underlying table to hold the search text
Add a text input component to the screen, and target it at the user specific column
Use a single value column to apply the search text to all rows
Create a filter on your inline list that compares the entered text to whichever column you want to search on
Once you have the filter in place, you can create a custom action on your inline list that first clears the search input, and then does a show details for the item selected. One point to note is that because each list item is attached to a separate row, you would need to create a single relation to link each row to the first row, and then do a set column values via that relation to clear the user specific column.