Is this two separate tabs or two windows? Are they the same homepage? What is your homepage?

https://support.mozilla.org/en-US/kb/how-to-set-the-home-page

Type about:config<enter> in the address bar. If a warning screen comes up, press the I Accept the Risk button. At the top of the screen is a search bar. Type browser.startup.homepage

What is the value? Right-click and select Reset to restore its default value; about:home

Thanks FredMcD for responding. In Preferences - Home - New Windows and tabs - the homepageand new windows is set to 'Firefox home' and new tabs to 'Firefox home' The same thing occurs if I set the above two lines to 'Blank page' The windows has the words 'New tab' top left.

browser.startup.homepage has value ' about:blank' setting this to ' about:home' make no difference

FredMcD said

Is this two separate tabs or two windows?
Are they the same homepage?

Start Firefox in Safe Mode {web link}

A small dialog should appear. Click Start In Safe Mode (not Refresh). Is the problem still there?

Two web pages - the second arriving about 2 secs after first. The problem does not happen in safe mode

It could be the work of one of your add-ons, or it could be Hardware Acceleration. How to disable Hardware Acceleration {web link}

Type about:addons<enter> in the address bar to open your Add-ons Manager. Hot key; <Control> (Mac=<Command>) <Shift> A)

In the Add-ons Manager, on the left, select Extensions. Disable a few add-ons, then Restart Firefox.

Some added toolbar and anti-virus add-ons are known to cause Firefox issues. Disable All Of them.

If the problem continues, disable some more (restarting FF). Continue until the problem is gone. After, you know what group is causing the issue. Re-enable the last group ONE AT A TIME (restarting FF) until the problem returns.

Once you think you found the problem, disable that and re-enable all the others, then restart again. Let us know who the suspect is, detective.

I have checked and I do not have any Addons. Only Plugin is Video codec automatically installed by Mozilla

And Hardware Acceleration?

What is your primary home page link? Are there any links in it?

Not sure what you mean by homepage links etc. Starting computer gives normal launcher column on left of screen which includes firefox. No acceleration Thanks for pursuing this - am fast suspecting a bug.

Let’s do a full clean re-install;

Note: Firefox comes in three or more folders on all computers. They are;

Maintenance: (Programs Folder) <Windows Only> Firefox itself: (Programs Folder) And two folders in the profile of each user on the computer for each Firefox profile for that user.

If you remove the Firefox folder, the user profiles would not be affected.

Download Firefox For All languages And Systems {web link}

Save the file. Then Close Firefox.

Using your file browser, open the Programs Folder on your computer.

Windows: C:\Program Files C:\Program Files (x86) Note: Check Both Folders

Mac: Open the "Applications" folder. https://support.mozilla.org/en-US/kb/how-download-and-install-firefox-mac

Linux: Check your user manual. If you installed Firefox with the distro-based package manager, you should use the same way to uninstall it. See Install Firefox on Linux; https://support.mozilla.org/en-US/kb/install-firefox-linux

If you downloaded and installed the binary package from the Firefox download page, simply remove the folder Firefox in your home directory. http://www.mozilla.org/firefox#desktop ++++++++++++++++++++++++++++ Look for, and remove any Mozilla or Firefox program folders. Do not remove the Mozilla Thunderbird folder if there is one.

Do Not remove any profile folders.

After rebooting the computer, run a registry scanner if you have one. Then run the installer. +++++++++++++++++++++++++++ If there is a problem, start your Computer in safe mode and try again.

http://encyclopedia2.thefreedictionary.com/Linux+Safe+Mode This shows how to Start all Computers in Safe Mode; Free Online Encyclopedia

I have just used the 'updater' to get the latest version of Firefox 68.0.2 (64 bit) - the previous version I was using was 68.0.1. Lo and behold only one window is appearing! So I'll mark my post as solved. Thanks for the comprehensive new install method and will make a record of it for use if the problem occurs again.

That was very good work. Well done.

Postscript. I have the found the cause of two windows appearing - it has nothing to do with 'Upgrading' after all.

On my startup screen when I turn on the laptop I have a number of shortcut files - to active these I need to 'double click' on them.

But Firefox on the side bar Launcher only requires 'one click' so a 'double click' will produce two windows!

As simple as that - but double clicking is so ingrained that I still make the mistake. Why there is a difference perhaps someone could enlighten me.