This is what I have done in a similar situation in the past.
Move the search icon to the left of the textbox and introduce an action which explicitly says search and below is the reasoning for my advice.
The search box is empty, not touched yet
In this scenario the search icon acts like a label for the textbox which means on click of the button the textbox is in focus and the users can start typing into the textbox
The search box having some results
In this case if the search button has been clicked the user seems to be looking for something that the user is unable to find in the instant search results. I have worked on a booking engine in the past and in our scenario we had instant search results which included property names along with destinations and the results were capped at 10. On selecting a result the user would be taken to the property or the destination page. In addition we also had an explicit search button build in to avoid a dead-end for the users if the results they are looking for are not part of the result set. If possible measure the clicks on this search button to see how many such clicks happen if they happen to be a lot then it gives you an indication to tune your search algo.
Hope this helps. Happy to chat further if required!