In a large pot, add the potatoes and cover them with cool water.
Bring the water to a boil and cook the potatoes until they are fork-tender.
Once the potatoes are cooked, drain them and transfer them to a large bowl.
In a medium bowl, mix together the hard-boiled eggs, mayonnaise, relish, mustard, red onion, celery, garlic powder, cayenne, paprika, and salt.
In a small bowl, stir together the vinegar, salt, and sugar to create the vinegar mixture.
Pour the hot vinegar mixture over the cooked potatoes and let them sit for 5 minutes to absorb the flavors.
Add the egg mixture to the potatoes, tossing to coat them completely.
Let the potato salad sit for another 5 minutes before serving.