gmail password in Mail no longer works

All of a sudden, my Mail program won't receive or send emails, by gmail (and it's a POP account, not IMAP). It appears that the passwords have got out of sync somehow, and I can't fix it.

First, I did an account recovery for my Google account. Then I changed the password on the Google account. Both were apparently successful, but I'm not sure whether these actions are for gmail or just the online Google account page.

Then I went into Mac Mail 'Settings' and selected 'server settings'. This gave me two locations where I could edit my passwords (one incoming and one outgoing). So I put in the newly minted password I had just done for the Google account. I got a message saying "Unable to verify account name or password".

I'm still not getting any incoming emails and I can't send.

Some additional points>

1. I can still log into my Google account, and see my incoming emails there, so all isn't lost.
2. This all started on the first day of a trip away from home, when trying to access gmail from an iPad. Possible finger trouble there by the user.
3. I can't find my gmail account password in Keychain - I'm sure it was there once, so how do I find it?
4. The gmail account is used on a Mac, an iPad, and two iPhones. Same problem on all four.

How can I find out what passwords gmail is expecting, so that I can edit my Mail server settings to match? Or find a way to change the gmail password in Google? Or is there another way to get these back in sync.