The sentence states that the mayor "easily relinquished her already minimal resolve to avoid confrontation." This indicates a lack of courage or a tendency to back down.

We are looking for words that describe someone who is easily swayed or lacking in courage, especially in the context of avoiding confrontation.

Let's evaluate the options:
- A. temperate: Showing moderation or self-restraint. This doesn't fit the idea of easily relinquishing resolve to avoid confrontation.
- B. craven: Lacking in courage; cowardly. This aligns perfectly with "easily relinquished her already minimal resolve to avoid confrontation."
- C. timid: Showing a lack of courage or confidence; easily frightened. This also aligns perfectly with "easily relinquished her already minimal resolve to avoid confrontation."
- D. moderate: Average in amount, intensity, quality, or degree; not extreme. This describes a middle-of-the-road stance, not necessarily a lack of resolve.
- E. self-serving: Having concern only for one's own interests. While possibly a reason for avoiding confrontation, it doesn't directly describe the stance as being weak.
- F. radical: Advocating for thorough or complete political or social reform; extreme. This is the opposite of someone who easily gives in.

Both B. craven and C. timid accurately describe a stance that lacks courage and is easily given up to avoid confrontation.

The final answer is $\(\mathrm{B}, \mathrm{C}\)$