Cooking a mix of raw and cooked meat
Once the meat is cooked, there really isn't a way to turn it into minced meat without there being a noticeable difference in texture. However, it is not unusual to see fully cooked meat getting finely chopped (in fact) to become, for example, filling, and used in ways sort of like how one would use fried minced meat.
Also, from The BBQ BRETHREN FORUMS:

You can grind cooked meat, no problem. Some sausage recipes, actually call for ground cooked meat.

You best bet is to cook the raw minced meat as you would usually cook it, and dump in the cooked meat, that's all chopped up, at the last moment to heat it up.