Thanks for the hints. I am now again in the live system trying to fix it. So I did:
sudo mount /dev/sdb5 /mnt
sudo mhwd-chroot /mnt
Another Terminal window pops up: root@manjaro:/ and says:
So I am trying here to
[manjaro /]# sudo pacman -Syyu
But I am wondering that I am not asked to login to my existing installation. Do I not need to enter my password for the existing installation as I have to do it, when doing something using root?