We can tell from the context of the sentence that the word for the blank needs to mean something similar to answer ambiguously. Because of that, the answer is "equivocate," which means use ambiguous language so as to conceal the truth or avoid committing oneself.

We know that whatever the squire did to the shield resulted in it being very clean and shiny enough for him to see his reflection in it. Given this, the correct answer is "burnished," or polished (something, usually metallic) by rubbing.

The sentence pairs the missing word with the subject's tendency to "kiss up" to his boss. This means the correct answer will mean something quite similar to "kiss up." "Fawn," meaning to behave sycophantically or grovel, is the best answer choice.

The action being done by the water is taken towards drought conditions. As a drought is a lack of water, this means the correct answer will mean helping or making less severe, the definition of "alleviate."

The "gummable bits" mentioned at the end of the last sentence provides the best clue to the answer. The action taken by the missing word must produce the small bites a "toothless dog" can eat. "Minced," meaning chopped into very small pieces, is the correct answer.

Since it appears the couple "seemed to dislike each other," it is safe to assume the missing word indicates some kind of disagreement. The fact it is over "the most minor things" also helps lead to the correct answer. "Quibbling," meaning arguing about small matters, is the best answer choice.

Since the subject is a "criminal" who is trying to "fool an undercover policeman," it is safe to assume the missing word has a negative connotation. The correct answer should mean scheming or tricking, the definition of "conniving."

Since the general "did nothing" after promising "to change the schedule, the soldiers were clearly annoyed by his shift. The correct answer will indicate the general changed his approach. "Reneged," meaning to go back on or retract a statement, is the best choice among the answers.

The "accused criminal" is trying to do something regarding his crimes with "community service." The use of "accused" indicates the subject is trying to do something positive about his crimes. "Expiate," meaning to atone or make amends, is the correct answer choice.

The sentence explicitly says the professor lectures on "whatever she felt like discussing."  This indicates the "assigned reading and course description" had very little bearing on the subject of the lectures.  The sentence's use of "did not usually," however, means the correct answer needs to mean be relevant to or address, the definition of "pertain."