Time. You can try adding time.
So, one of the things mixing really needs to do is equalize the moisture - mix the dry into the wet ingredients, so the batter is (mostly) uniform instead of pockets of drier flour in swimming in liquid. Mixing the ingredients about does this quite well, but it can also develop gluten if you mix it too much or too fiercely, which seems to be the problem you're having.
Letting the wet ingredients sit with the dry should let everything get moist and hydrated before you finish mixing it, and so the same amount of mixing should end up with a more uniform product - especially if you stir together roughly, let it sit for the lumps to moisten, then finish mixing into a fairly smooth batter.
It is also a lot easier to mix ingredients if everything is closer to the same consistency, like adding two dry or two moist ingredients together - and after absorbing a bit, the flour should be looser and moister, the liquids should be thicker because some has absorbed into the flour, and everything should stir together much more smoothly instead of sliding past each other. You may still get little lumps, but they should be little wet lumps of batter that will smooth out with heat and time while baking, not big pockets of dry flour that would leave the finished cake lumpy and wet and undermixed... and certainly not overdeveloped gluten to make the cake tough.
It's worth noting that gluten is not the enemy, by the way. It gives some structure and helps catch the bubbles that make your cake rise. You don't want a lot, granted, if you want a cake that's soft instead of chewy - but if having none was better, cakes made with actual gluten-free flours would be so much more popular than (wheat) cake flour. Also, it actually takes some doing to develop serious gluten (lots of proteins sliding past each other and snagging), so smoothly mixing your batter to a mostly uniform consistency and avoiding lumpy cake shouldn't be a problem, spending a lot of extra mixing time furiously trying to smooth out every little lump may be too much.