Whisk together mayonnaise, relish, lemon juice, 1 tablespoon of the dill, and 1/4 teaspoon each of the salt and pepper.

Whisk together cornmeal, flour, and remaining 1/2 teaspoon each salt and pepper in a large bowl. Make a well in the center; add buttermilk, clam juice, and eggs, and whisk just until combined. Fold in chopped clams, onion, corn, and remaining 2 tablespoons dill.

Pour oil to a depth of 3 inches in a large Dutch oven; heat oil over medium-high to 350°F.

Working in batches, carefully scoop heaping tablespoons of batter into hot oil, and fry, turning occasionally to cook evenly, until deep golden brown, about 3 minutes. (Take care not to overcrowd the Dutch oven.) Transfer to a plate lined with paper towels to drain. Serve immediately.