I tried the repair option, but, as you surmised, it didn't solve the problem. I then tried a complete reinstall of Avast Premium Security. The re-install worked fine with it finding my existing Activation Code It asked me to sign in and I did without any problems using my existing password. Then I tried to go to some 'protected areas' including 'Settings'. In every case it would not accept my password! So I still have the problem, even after a complete re-install.
My wife has Avast Premium on her computer; it is one of, (actually the only), device hanging on my multi-device account. She can get into Settings without her being asked for a password at all... it just goes straight in. Her computer is virtually identical to mine, an HP All-in-One desktop with the same version of Win10 and the same version of Firefox browser.
We both have Malwarebytes Pro and after I re-installed Avast Premium the first thing it did was tell me Malwarebytes should be uninstalled as they 'might get in each other's way'. I haven't done that yet, but am prepared to if it will solve the problem, but having said that, it hasn't caused any conflict on my wife's machine. We initially installed Malwarebytes as it covered areas Avast did not at that time. With ongoing improvements to Avast perhaps we no longer need Malwarebytes anyway?
What can I do now? I have followed all your suggestions. Do you have any other thoughts? I am in Australia so can't ring Avast's US 1800 phone number. Is there another way I can ask their technical department direct?