You can say either 'correct a mistake' or 'fix a mistake'. As John has explained, 'fix' means to correct something which isn't working. If a student makes mistakes in their writing by using the wrong grammar or vocabulary, you could say that they need to 'fix the mistakes' by replacing the wrong words with the right ones. On the subject of choosing words.... I admire your ambition to expand your English vocabulary, but 'oscillate' isn't quite the right word here. Things oscillate ( pendulums, temperatures and so on), not people. I'd say something simpler, along the lines of "I can't decide between x and y".

John..Thank you very much, that's precisely what I expected. Fairly relevant and detailed explanation..

correct is correct, to fix is used for repairing broken things or bad situations like a relationship or marriage that is not working. Fix = to make work/operate correctly something that is broken. To correct = to make something how it should be, before going off course/track. Originally the word "to correct" was only used to refer to correcting navigation to put a vessel back on the right course. You correct your steering on a car to avoid collisions. Hope this helps to explain,