OFFICIAL EXPLANATION
Jane was with the impression that John needed her help.(A) with the impression that John needed her help
This answer choice is grammatically incorrect. The correct idiom is [someone] is under the impression.(B) with the impression that John could use her help
This answer choice repeats the original Preposition mistake. The correct idiom is [someone] is under the impression.(C) under the impression that John needed her to help him
This answer choice corrects the original Preposition mistake by using the correct idiom [someone] is under the impression.(D) under the impression that John could use some help from her
Although this answer choice corrects the Preposition error in the original sentence, it is stylistically flawed as it uses the redundant phrasing could use some help from her.
There is another grammatically correct answer choice which is more concise. Look for it!(E) under the impression that John needed help
Although this answer choice corrects the Preposition error in the original sentence, it changes its meaning by omitting the word her. In the original sentence, John specifically needs Jane's help whereas in this answer choice he needs help in general.