According to the Help Center, A book I sent was returned to me!, I am supposed to send a PM to the requestor to verify their address and ask if they still want the book. However, I am not willing to pay postage 2-3 times for only 1 credit. If the requestor still wants the book, I feel that I should receive another credit for re-mailing the book. Is this possible? Would it be wrong for me to ask the requestor for another credit?

I am aware that I am not obligated to re-send the book and that I can simply repost it. However, I would like to give the first requestor another opportunity to receive the book, as long as they give me a credit or otherwise pay for postage. I just don't know if that is acceptable.

If I simply repost it, should I do anything with the current transaction? Is there any way for me to cancel it, or do I just let it get "lost"?

First  you need to contact the requestor and verify that that was indeed the correct address.  If it was the correct address and the USPS made the error they will typically resend the package for you without additional cost....except if you didn't put enough postage on it you will probably have to add that to it.  If the additional postage  you are talking about is for the USPS returning the package, you will owe that IF indeed the address was incorrect.  Book Rate postage is only a one way postage, it does not cover return of the book.

All that said...First you need to contact the sender and figure out what happened....

1.   was the address correct or incorrect.....

This needs to be done before you can make any other decisions.....

then per the help center if the requestor did not update their mailing address or typed it in wrong:

You are not obligated to re-send the book if this happened due to a problem with the requestor's address.   If the book was not damaged when it came back to you, and you do not want to re-send the book to the previous requestor, you can simply repost it.

When you contact the person, be sure to give a deadline for response. PBS typically recommends 7 days. That way, you can have a clear-cut end to the transaction. Without the deadline, you might be stuck waiting and wondering.

I think in this case the Help Desk recommends 5 days....I THINK I read that this morning but, I need a nap now so, I could have been dreaming...