Official Explanation
A question about the Henry V's victory at the Battle of Agincourt in 1415.
Split #1: As explained in this blog, this question contains a "Case I" use of the "with" + [noun] + [participle] structure, action by a different actor. In the main clause, the English longbowmen are the actors, and in the "with" phrase, the muddy field is another actor. This is wrong. One way to see this, as discussed above, is to examine the sentence without the participle:
With the muddy field, the English longbowmen at the Battle of Agincourt were able to inflict significant damage …
This makes no sense. The English longbowmen weren't doing anything with the muddy field. An essential piece of the action has been lost in dropping that participle. For a separate actor performing a separate action, we need a whole new clause. (A) is wrong because of this.
The sentence is radically reorganized on each choice, so we have to analyze each choice separately.
(A) In addition to the problematic structure at the beginning, this choice repeats the Case I mistake at the end, a double whammy! This is completely incorrect.
(B) This choice has the Case I mistake after the word "Agincourt": an "with" + [noun] + [participle] used incorrectly. Also, the antecedent of "their" is exceptionally unclear and ambiguous. This choice is incorrect.
(C) This choice makes no mistakes. All the pronouns have clear relationships with their antecedents. The logical connections are clear. This is promising.
(D) This choice has a combination of small problems. The tense mismatch in the first part, "tried … but had been delayed," doesn't make sense. Presumably, the "trying" and the "being delayed" would be more or less simultaneous. Also, that first part is very wordy. The overall sentence is just connect with a string of and's, and this gives no sense of the logical interconnection of these events, what cause what. The antecedent of "their" is ambiguous. The hypothetical tense, "would eliminate," is strange and awkward for a factual historical event. This choice is incorrect.
(E) This choice also links the information with "and" statements, which make the logical sequence less obvious. The antecedent of "them" is not perfectly clear. The passive structure at the beginning is weak. This choice is grammatically correct but far from ideal.
The best answer is (C).