"The mistakes are found to be corrected." You cannot say "to be corrected" in this sentence. You should say "The mistakes are found to have been corrected." Even this sounds odd, and very formal, but is it grammatically correct. "The mistakes are found correct." This sentence should read "The mistakes have been found to be correct." The second set of sentences are both correct.

The mistakes are found to be corrected. - nothing wrong here, not sure what you're asking "to have been corrected" might be better but depends on the context. The mistakes are found correct - the meaning is different from the previous sentence but nothing wrong grammatically either but whether it makes sense is questionable. I want my English to be imrpoved. - nothing wrong too. "to be improved" by some practice. I want my English to improve. - means to become better regardless of practice or at least the sentence doesn't focus on its presence.