Of late some of my favourites open, accept my login and PW but do not open the subsequent page. No attempt just “the page cannot be displayed”. Typing in the address and bypassing favourites does not help.
Problems may have started since my last Java update, so it is a suspect.

Any ideas?
Sony Vaio with XP fully updated, IE

Have your security settings in IE changed/been changed?

Try using another browser, like Firefox, you may never want to use IE again!

https sites? Clearing the SSL cache might help:
http://support.microsoft.com/?kbid=813444

Thanks DBTL. That link headed me in the right direction and tweaking Norton Firewall settings recovered all sites but 1. My most mportant site where I read my work emails but it does not have a https prefix.

Strange indeed. There still might be a single security setting of your browser that's not what it should be. Firewall to my logic is not the culprit.
There's a trick that might help in the troubleshooting. Google for "anynymous proxies". They are free services that redirect you from their site to the site you actually want to visit. For some services, some popular sites are blocked and not everything is supported, like scripting. See if you can reach your site through any of them.

The latest Windows update, MS06-013 could be the root cause as you suspected:
"Minor problems have also been reported with early versions of Google's Toolbar and some versions of Sun's Java", moreover: "This patch changes the way web browsers interact with ActiveX controls". At least, downloading the latest Sun Java should not be harmful:
http://www.java.com/en/download/manual.jsp