In a slow cooker, combine the first four ingredients. While stirring, pour in the polenta in a slow, steady stream.
Cover and cook on high for 1 1/2 to 2 1/2 hours or low for 3 1/2 to 4 hours. There is no need to stir during this time.
About 20 minutes before the polenta is done, heat the oil in a large skillet over medium-high. Add the paprika and cumin and sauté for 30 seconds. Add the turkey and sauté, breaking up any large clumps, until the meat starts to brown, about 6 minutes. Add the onion, red pepper, and corn, and sauté until soft, about 7 minutes. Stir in 1/4 cup of water to deglaze the pan, then stir in the barbecue sauce. Season with salt and pepper as needed. Cover and set aside.
Stir the cheese into the polenta, then taste and season with salt and pepper, if needed.
To serve, spoon the polenta onto plates and top with the turkey mixture.