Can the self correction mechanisms in cells imply that the DNA code to a degree is self modifying

There is an argument that proteins coded by 20 amino acids via the DNA program has so few successful protein formation that a search never can reach the complexity of it via natural selection and genetics. This is a very similar argument as that of the impossibility of the eye. Again the answer is probably that the advanced proteins evolved from other proteins or from proteins that was smaller and now extinct. But still there is another theoretical possibility that I find interesting. As I understand the cell can correct mutations, but if the DNA can encode how to correct we get a self modifying code at least to a degree that can tune and change the mutations to more successful protein mutations. As a fan of Allan Turing and his turing machine, that would be a cool feature of the genetics in the cell. Is there any proofs or indications of mutation cooprated self modifications of the DNA program?