AzDude & Basj - thanks to both for your help. I had a look at the webpage Basj recommended and it lead me to the answer, though I think their explanation isn't perfect or universal.
They say "... and in most cases they’re located at www.domain.com/favicon.ico. " I suggest that's misleading and NOT the immediate answer in every case. If you go to SOME websites and add /favicon.ico after the domain name, that will indeed take you to a page with the icon, from where you can save it off to your preferred location.
I've tried that with a few websites. It worked with the specific site that prompted me to ask the question, so that's now sorted - again, thanks. It also worked with some others but not with all.
My initial guess is that it may depend on how "well" the site itself is written. For example, on a club website that I know is not professionally set up it doesn't work.
Anyway, it's been something I've been meaning to resolve for a while - hope it helps others.
