How to properly prep for new backsplash?

Hello! I am installing a new backsplash in our kitchen. I've done it twice before but both times required minimal wall prep because it was on fresh drywall. This time, I tore out the old 'backsplash'. It was just some thin wood paneling attached to drywall with some adhesive and Brad nails. I am ready to tile but need to fill a few holes left behind when I took out the old paneling. What should I use to patch the holes that will be sturdy enough for my backsplash to go over? Patching plaster? How much of the old adhesive do I need to scrape away?
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