For some time my macbook air fails to update to 10.15.4, with no error message given.

Good question! It made me look again, as I questioned whether I had written down 10.15.4 in error. But no, I'm running 10.15.3 and I get the standard message to update to 10.15.4. Now I have just tried again - with the same result, but after trying I still had the 10.15.4 message, which changed a few seconds later to 10.15.6! So I tried that update, but again with no luck - back to the 10.15.4 message! My mac appears to be as confused as I am! I'll now try the Combo link you've suggested.

Thanks for the suggestions. I've tried both updates, including the Combi versions, but without any success. After each attempt my mac comes back to the logon screen after about 15 seconds, always without any error message.

Thanks to both of you for your guidance. I'd been having free space problems for some time since I lost control of iPhotos and iCloud. With only 120GB, it became critical. I regained control and the current problem of free space has been solved by deleting an old iPhone backup no longer needed, and by transferring my current iPhone and iPad backups to another machine.

Catalina 10.15.6 is installed and I now have 34GB of free space.

I have another problem to resolve, relating to the appearance of the Recovery Assistant on both my machines, nearly every time I boot up, but before the logon screen. I'll start a new discussion for this one, so please watch out for it.

Thanks again.