In a large bowl, stir together the first four ingredients until the syrup or honey has dissolved. Add the yeast and stir to combine. Let the mixture sit until it's foaming and bubbly, about 5 minutes. Stir in the flour until a soft dough forms.
Sprinkle a dry work surface with flour and dust your hands as well. Turn out the dough and knead it: use the heels of your hands to press and flatten the dough, fold it onto itself, and give it a quarter turn. Continue to knead, fold, and turn until the dough is smooth and elastic, about 10 minutes. If the dough is too sticky, add a sprinkle of flour as you work.
Oil a large bowl. Place the dough inside and turn it to coat it with the oil. Cover the bowl with a clean kitchen towel and set it in a warm spot to rise until doubled in bulk, about 1 1/2 hours.
Line a large, rimmed baking sheet with parchment paper. Heat the oven to 450° and place a rack in the center position. Fill a large pot with at least 3 inches of water. Stir in the baking soda and bring the mixture to a boil.
Meanwhile, punch down the dough. Sprinkle your work surface and hands with flour and knead in the apple and cinnamon. Separate the dough into eight equal portions. Roll and shape the dough into pretzels then place them on a platter.
Bring the water to a simmer and drop two pretzels into the pot (they should float freely and not touch). Boil them for 2 minutes, turning once. Use a slotted spoon to remove them from the water and place them on the prepared baking sheet, knot side up. Repeat with the remaining pretzels.
In a small bowl, whisk together the egg white and 2 tablespoons cool water. Brush the egg wash on the pretzels, then sprinkle them with the sugar.
Bake the pretzels, turning the pan halfway through, until they're puffed and browned, about 12 minutes. Let them cool at least 15 minutes before serving.