Drain the tuna thoroughly, pressing the lid down and then pressing again with a fork in the bowl. Drain the relish against the side of the jar.
In a medium bowl, mix the tuna, mayonnaise, onion, relish, Dijon, garlic powder, and black pepper until creamy but still flaky.
Stack the tortillas on a plate, cover with a damp paper towel, and microwave 20 to 30 seconds until warm and pliable.
Sprinkle a small handful of cheddar down the center of each tortilla. Divide the tuna salad between them in a line on top of the cheese, leaving 2 inches clear at each end. Top with the remaining cheddar.
Fold the two short ends in, then roll tightly from the long edge like a burrito.
Brush all over with melted butter, including the ends.
Preheat the air fryer to 380°F. Place the wraps seam side down in the basket with space between them. Cook 6 to 7 minutes, without flipping, until deep golden and blistered.
Rest 2 minutes before cutting so the cheese can set.