I know that both "get it right" and "make it right" have a meaning of "correction", but what's the significant difference between them?
The two phrases can be used in different contexts, so it'll be hard to explain ONE difference that always holds true.
That said, I'd suggest that:
As I said, though, that's only one way to distinguish the two phrases. A math teacher might tell her students:
But then, if the students turn in answers with a lot of mistakes:
Essentially, make it right refers to fixing something your got wrong, whereas get it right is an exhortation to not get it wrong in the first place.
"Get it right" means that you are going to do something right the first time. You're working on a project, and you are determined to do it the correct way.
"Make it right" means more to fix something that is already wrong. You've taken something already created (or some harm already done) and decided to fix it. If it's so bad that it can't be repaired, you might scrap it and start over, planning to "get it right" the next time around.
"Make it right" also has more of a moral connotation. To "make something right" most often means to make amends for some wrong that has been done to someone. "He had his TV stolen from his house, but the cops will catch the thief and make this right."
In 'making it right' you are asking a person to correct an injustice, to change a state from being unfair to fair. The person who is being asked obviously has some ability, power or influence to be able to alter whichever situation you are trying to make fair again while the person asking lacks or is unable or incapable of changing the unfair situation themself. "Honey, please make it right between your parents". "You are our last hope, you have to make it right no matter the sacrifice".
With 'get it right' you are asking someone to succeed during their next attempt, which could be the first attempt, but could also be additional attempt at achieving something they failed at achieving before. It is said from someone in a position of power to someone below them, since this is not something a soldier would ever say to a general, an employee to a boss, or a child to their parents (unless they got no manners).
A coach, a teacher, a parent, a team mate, or anyone with a stake in another person succeeding at some task is likely to say this as long as they are above or equal to them. Saying 'get it right' will sound unusually harsh coming from a spouse, or close friend if used incorrectly, and for some reason I want to add 'bozo' to the end - "get it right bozo... or you are fired!" "You will keep marching around the barracks all night and every night, if that's what it takes for you to get it right."
So "make it right" is addressed to a position of power in a polite, almost pleading way, hoping that they will accept. While "get it right" is never addressed to a position of power unless you want to sound insolent and rude on purpose.
Make it right means something or some situation is wrong and needs correction. Get it right means doing it right or the right thing in the first attempt. No room for doing wrong .or It being done wrong ..