obviously I don't want to have an account which is vulnerable. What should I do?

This is Google speak. Your account is by no means 'vulnerable', it just doesn't use the Google preferred authentication method right now.

To fix that set Authentication type to 'OAuth2' in your Gmail account settings for the IMAP server settings. Do the same for the Outgoing Server (SMTP).

Then log in using your Gmail password.

Wrt to Googlemail vs. Gmail, it is both the same. If you have set up your account as googlemail.com, just leave it as it is.

Thank you very much Christ1, now I know not to worry about that message and googlemail v gmail. However I still have problems with the imap issue. Here's what I did and what happened (I haven't found a way to change the smtp, but since the password request comes from imap, this is what I concentrated on) - I surprised myself by finding where to change the imap authentication to OAuth2 in the Googlemail settings, done. - got a Sign in with Google Account, pressed Next, input my (new since yesterday) password. - got the Accounts Help page, with a little Sign in box in the top right, tried that (the top bar shows ' Enter credentials for googlemail.com on imap.googlemail.com') - entered googlemail address and password, got to same Accounts Help page with Sign in at the top right. - looked at the Troubleshoot column on the right, clicked Passwords, found nothing useful - typed 'Enter credentials for googlemail.com on imap.googlemail.com' in the Search box, clicked Search, nothing happened. So, what am I doing wrong, do you think? Any ideas? Your help is greatly appreciated by the way, I feel less lonely trying to sort this mess... PS I can send screenshots if you wish me to.

A screenshot would probably help. https://support.mozilla.org/kb/how-do-i-create-screenshot-my-problem

Sorry, I missed your reply (didn't receive it in my email for some reason), I'll see what I can do. Many thanks... I have made 3 screenshots: 1. My Thunderbird/Googlemail account setting with Security settings showing the imap options - when I try to connect with googlemail with your imap setting I get 'Failed to connect' 2 & 3 - A window which kept popping up yesterday when I was in Thunderbird (happening again today). The window shows what happens when I try to log in the 'Enter credentials... on imap... gmail.com). This comes up when I'm in Thunderbird and it looks like its googlemail's new way of asking me the imap password. Thunderbird was updated to v 45.5.0 on 1.12.16. It may have something to do with it? (Just thought of it as the Uber issue - see below - happened the following day) Please let me know if there are any screenshots which may be useful. Another issue has come up, similar to this: when I try to connect to Uber with my gmail address/password, I can't connect via my computer - although the app on my phone works fine. So I'm not at all sure where the fault might lie, and I'm really not very clued up techwise, as you've no doubt realised. With many thanks for your help and patience

Make sure you do allow cookies for https://accounts.google.com in Thunderbird.

Thank you Christ1, I must seem very dim to you, in fact I am. Where would I look in Thunderbird, to check if they are allowed or not? I looked in the Search box but all it does is show me Mac stuff, nothing specifically about Thunderbird... Thank you for your help.

At the top right of the Thunderbird window, click the menu button > Preferences -> Privacy

Leave 'Accept cookies' from sites' unchecked.

Click 'Exceptions' and add the above given URL for Google as an exception (Allow). This shouldn't be needed in the first place, but it will hopefully get you going.

'Accept cookies' is unchecked anyway. What above URL are you talking about (to use as exception)? Thanks

As said at https://support.mozilla.org/en-US/questions/1149153#answer-943046 :

Make sure you do allow cookies for https://accounts.google.com in Thunderbird.

Ok, I didn't realise you meant to allow this URL: https://accounts.google.com/ (I hope I've got it right this time). and I have added it as an exception. I'll have to see if that works. Does this trick override the need for an imap password then? Many thanks again.

Does this trick override the need for an imap password then?

Certainly not.

Ah ok, so what is it for? And how do I solve the issue that started everything, the imap password? How can I find out what it is, I've tried all the passewords I can think of. And I can't find a way to contact Thunderbird to ask them...

I have added it as an exception. I'll have to see if that works.

Please report back what happened.

how do I solve the issue that started everything, the imap password?

There is only one password - your Gmail password. If you changed your Gmail password recently on the Gmail website, this is the one you'll have to enter when being prompted for a password by Thunderbird.

I can't find a way to contact Thunderbird to ask them...

Do you mean to ask 'Thunderbird' what your password is? That won't work. First of all, Thunderbird is a piece of software to retrieve your mail, not an organization. You can tell Thunderbird to remember your password, so that you won't need to type it again and again, every time you start the application. Second, Thunderbird is a community project, not a commercial entity. This forum is the official support venue, there is no phone support for Thunderbird.