In California and New York, Democratic challengers will need to spend

Posting the original answer that was given.

This confusing problem is a perfect example of how important logical meaning is in sentence correction. On your first read through, the original might seem awkward or incorrect, but it is the only sentence that conveys a logical meaning and has a correct grammatical structure. The sentence in (A), after the proper slash-and-burn is really: “The democratic challengers will need to spend more heavily TO DO THIS OR TO DO THAT.” Importantly those two things (the THIS and THAT) are very different and unrelated to each other (beat back their own rivals OR introduce themselves to voters in order to compete against well known candidates of the other party). (B) contains a clear structure error - you cannot say “challengers will need to spend….., introduce” - some type of connector would be required. (D) also contains a structural error as you cannot say “challengers will need to spend more heavily…and introducing” In (C) and (E), the structural issue is corrected but the sentences are entirely illogical: They are not going to beat back rivals of their OWN party “by introducing” or “through the introduction of themselves” themselves to voters against well known Republicans. These are clearly two separate things and these structures cannot be used to convey a logical meaning. Only (A) is correct and contains a proper structure that conveys a logical meaning.