In the container of a food processor, combine the mayonnaise, mustard, horseradish, garlic, and lemon juice. Process until well blended. Transfer to a bowl, stir in chives, cover and refrigerate until ready to serve.
Heat the oil in a heavy saucepan or deep fryer to 375 degrees F (190 degrees C). Stir together the hush puppy mix, milk and egg until smooth. Fold in the crab. Drop tablespoonfuls of the batter into the hot oil. Fry until golden brown, 2 to 3 minutes, turning once if needed. Drain on paper towels. Serve fritters with the garlic sauce.