Most people call it punishment but i do think "correction" fits this much better. It is not really about "making a girl pay for her action" but more about improving her and making her realize her mistakes.
Corrections are supposed to be painful. They should hurt. And yes, even make the girl cry sometimes. Depending on the transgression of course. The "punishment" needs to fit the crime.
But what it really is about his correction and improvement. The Sir doesn't like to punish his girl, to hurt her, and to make her scream. Of course, there are sadists/masochists that do this as play - but play and correction are very different. A correction should never be enjoyed by the Sir. It should be just but not fun.
A correction helps the girl realize her mistakes. It makes her remember the mistakes she made and also makes her sorry for doing them. Saying "i am sorry" is very easy but feeling sorry can only be guaranteed by some just discipline. Even though many girls feel remorse at breaking rules or making mistakes on their own, without even a Sir reminding them of their transgression, you should never forgo a correction like.
Because corrections serve another purpose: making a clean slate again. We girls tend to overthink and usually have a million things on our minds. And making a mistake, breaking a rule, upsetting our Sir, etc. that just weighs very heavy on our conscience.
A correction fixes this by giving us a clean slate again. With repentance, we pay off our moral debt. After the correction, after we took the punishment, we are good girls again. That is also why it is very important for the Sir to never bring this incident up again. The girl paid her dues, she is a good girl again. That is why there can't be hidden hatred or ill thoughts because each mistake is dealt with. Of course, if a girl repeats the same mistakes the Sir can order harsher punishments to incentivize her to not do it again. But by making this clear what is good and what is bad behavior, everyone wins. The girl knows exactly what is allowed and what is not - and girls work best in fixed structures. And the Sir doesn't have to hide his disappointment in the girl, he can actively work on making her better.
In the end, everyone gets what they need: girls get structures and don't have to overthink too much because every mistake is atoned for. And the Man gets an obedient girl that behaves exactly the way he enjoys.