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Isn't cornbread, like cheesecake, horribly misnamed for what it is? IE, it's not a bread at all: it's a cake, just as cheesecake isn't a cake, it's a pie.
That said, treat it like a cake: use cake/soft flour because it has less gluten than AP/plain flour, making for a lighter, fluffier cornbread. Secondly, mix an acidulant with a base. This could be as simple as just using baking powder, which has both an acidulant and base in powdered form, but for best results, whisk bicarb/baking soda into the dry ingredients then add a sour liquid to the batter. Your recipe will most likely use dairy and making a sour liquid could just be adding a tablespoon or two mixed into it, such as vinegar or lemon juice. Some prefer whole/as-is sour liquid, to which nothing needs to be added, such as buttermilk or soured cream.
Finally:
Do.
Not.
Over.
Mix.
A few lumps in the batter are fine. Too much mixing, even when using cake flour, will promote glutenisation, yielding a chewier, tougher baked good. This is why people knead bread dough before baking: you generally want bread to have a little chewiness and spring to it, but you want cake to be light and fluffy.
So there you have it. Hope this helps and good luck in your baking.