I guess this is not, strictly speaking, a reply but I don't know where else to post it and it might be useful to someone. I had a brainwave and I copied and pasted the URL into I.E. and opened it there. After closing I was delighted to find a cookie in CCeaner which I transferred to the right-hand pane to save.I closed down my PC and when I rebooted, the cookie was saved in Firefox (though the History had been cleared)and I was able to visit the website. I've shut down twice since and each time the History and other cookies are cleared but the cookie I need is saved in Firefox - because it is in CCleaner. So full marks CCleaner! But if there is an IT genius reading this I would love to know why CCleaner doesn't "pick up" cookies from Firefox as it does from I.E. Firefox is widely used now so please - can anything be done. Even if not, many thanks for four fantastic little helpers - CCleaner, Defraggler, Speccy and Recuva. My PC had to be rebuilt last week and I couldn't find the ID for Windows installation. There it was in Speccy. A grand-daughter was distraught last summer when she lost four hours course-work required for A levels. I introduced her to Recuva and she recovered the lot. (It took a very long time)
As someone from Piriform says - Huge thanks.