First of all, we can state:
Of the combination 97360, 2 digits are in the right position, and 1 digit is in the incorrect position.
Also, 50928 has 1 digit in the right position, and 1 digit in the incorrect position.
When we look at the overlap in used digits between 97360 and 50928, we find that only 9 and 0 occur in both. Because the positions of these digits aren't the same for both, (1,5 and 2,3 respectively), and 50928 doesnt have "2 correct digits in the incorrect position", we can conclude that "9 AND 0" aren't both in the final code.
If either 9 or 0 is in the correct position in 97360, the condition for 50928 "one digit is in the incorrect position" is satisfied.
I feel like it's fair to say that 16374 doesn't have any numbers in the correct position. If it did have numbers in the correct position, the puzzle should've mentioned that.
Because the overlap in numbers between 97360 and 16374 are the numbers 3, 6, and 7, where the number 3 is on the third digit in both of the codes, we could rule out 3 as being "one of the numbers in the correct position" in 97360. I'm basing this on the assumption that the puzzle didnt provide information about any digit being in the correct position in 16374.
Because '3' is neither in a correct nor incorrect position in the final code, we can conclude that 3 is not in the final code.
Therefore, 37068 and 57063 are not possible final codes.
We can also conclude that of the digits 0,6,7 and 9, two are in the correct position and one is in an incorrect position, so one of those digits is not in the final code.
Looking at my earlier conclusion where either 9 OR 0 is in the final code, we can conclude that 6 and 7 are in the final code.
This would contradict the statement about 16374 though, meaning we can't assume that 16374 doesn't have any digits in the correct position.
I'm stuck here, this was just my line of thought regarding this puzzle..