In a medium bowl, combine the mayonnaise, ketchup, sweet pickle relish, yellow mustard, and white vinegar. Whisk until these base ingredients are well incorporated and smooth.
Add the smoked paprika, garlic powder, onion powder, and cayenne pepper (if using) to the bowl. Continue whisking until all the spices are evenly distributed throughout the sauce.
Taste the sauce and season with salt and freshly ground black pepper as needed. Adjust the tanginess or sweetness by adding a tiny bit more vinegar or relish if desired.
Once you're happy with the flavor, transfer the burger sauce to an airtight container. For the best flavor, let it chill in the refrigerator for at least 30 minutes before serving to allow the flavors to meld.