Yes, moderator can remove bounty notice and refund the bounty.
According to Who are the diamond moderators, and what is their role?
- They can (at their discretion) refund and cancel a bounty.
Moderators can't edit the bounty message. So if you really need to edit the message then the only way is to flag and ask any moderator as they can remove and refund bounty and then you can start a new one.
 Note that removing or refunding bounty by moderator is only carried out in special circumstances i.e., closing, migrating, deleting featured questions etc. So, you are supposed to flag for removal of bounty for exceptional reasons only.
Related feature-request: Can we have a way to edit bounty custom message?
For complete information on bounty system, visit How does the bounty system work?
(see the related edit to the previous post)
Another question: On this site, if bounty is not having the desired effect of attracting more authentic answers, can moderators extend the bounty by one or two weeks? I.e., refund points and have user start a new bounty for another week without losing points or reputation?
No, moderators are not supposed to do anything in that case and bounty doesn't guarantee a desired effect or response.
From mentioned FAQ:
What happens if there's no answer after the bounty period?
If after the end of the bounty period a question has no answers, no bounty will be awarded and the question will no longer be featured.
Bounties are best understood as exchanging reputation for higher question visibility and increased answerer motivation. A bounty does not guarantee a response and is not refunded if none are received.
If you want to extend the bounty, you've to apply/start it again with double amount of reputation.