What is your impression of churchill’s sentence? does it seem correct or incorrect? does it sound like a good sentence or not? what might you change to improve it and why?

I believe the 'sentence' meaning in this context is the 'phrase' one, yes? As Winston Churchil was well known for his notorius quotes, and one in particular he stated to satirize a writing myth in English, usually understood as a rule, when criticized about doing it.  Which I believe is the one the question is refering to.

The writing 'rule' (myth) Churchill's reply satirizes is the 'Never end a sentence on a preposition' rule (i.g. as I intetionally did on the immediate sentence before this one). And his reply to it was something like 'This is the type of errant pedantry up with which I will not put.'

The 'rule' is a myth, yes, but of course what Churchill did was an exaggeration to sneeringly point out the ignorance of those who criticized him.

His sentence therefore was incorrect. One possible change to improve it could be: 'This is the type of errant pedantry which I will not put up with.'

Specially the 'up' and 'with' of 'put up with' could never go in the middle of a sentence, as 'put up with' is a phrasal verb, meaning the verb and the preposition must always be together in the correct order.

I was able to find some possible variations of what his sentence could have actually had been, but in none of them the 'up with' goes along with 'put', so either ways we can assume that his sentence was deliberately wrong.

It sounds pretty horrible. For one thing, up is part of the verb put up, so ending the sentence with just put seems distorted. If the sentence only used up at the end and not with, it would still sound odd: "Ending a sentence with a preposition is not something with which I would put up." It seems that the words "put up with" all go together, and splitting them distorts the sentence. So it would be better to break the so-called rule: "Ending a sentence with a preposition is something that I will not put up with." Or you could just use the word tolerate for "put up with" and not have a problem with prepositions. You could also avoid the "is not something" phrasing and just say "I will not tolerate (or put up with) ending a sentence with a preposition." (Humorously, it now ends with "a preposition" but not with an example of a preposition.)

Clearly Churchill wrote the sentence this way to make fun of people who will go to great lengths to avoid breaking a rule that doesn't really have a lot of credibility. I wouldn't really change it at all because it makes the point very well that following this rule would often result in ridiculous sentences.