Although I do have a backed up wallet file, I seem to also have a second bitcoin receiving address, which doesn't seem to correspond to it. Is there a way to recover any coins associated with that receiving address, or are the coins lost forever?
You can't get your bitcoins with just the receiving address (if that could be done, they could easily be stolen), so your only chances of retrieving them are:
- If they're stored in the wallet file you have backed up, then you can use that
- Somehow recovering the original wallet file, which is a data recovery exercise. This might be possible in the case of a failure or deletion of a hard disk, or a deletion from an SD card/memory stick
It sounds like you have an old wallet backup that does not have the private key to the public address you are looking for. You are probably in the realm of data recovery.
I wrote a blog article that might help: http://gary-rowe.com/agilestack/2012/08/17/how-to-recover-your-bitcoins-from-a-failed-hard-drive/