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Later I tried the "manual installation" at the github page... Failed in the same way:

mike@M17A ~ $  VERSION=node_10.16.3
mike@M17A ~ $  DISTRO="$(lsb_release -s -c)"
mike@M17A ~ $  echo "deb https://deb.nodesource.com/$VERSION $DISTRO main" | sudo tee /etc/apt/sources.list.d/nodesource.list
deb https://deb.nodesource.com/node_10.16.3 sylvia main
mike@M17A ~ $  echo "deb-src https://deb.nodesource.com/$VERSION $DISTRO main" | sudo tee -a /etc/apt/sources.list.d/nodesource.list
deb-src https://deb.nodesource.com/node_10.16.3 sylvia main
mike@M17A ~ $  sudo apt-get update
Hit:1 http://ppa.launchpad.net/adabbas/1stppa/ubuntu xenial InRelease
...

Ign:57 https://deb.opera.com/opera-stable stable/non-free Translation-en
Err:58 https://deb.opera.com/opera-stable stable/non-free amd64 Packages
  503  Backend unavailable, connection timeout
Ign:59 https://deb.opera.com/opera-stable stable/non-free i386 Packages
Fetched 3,316 B in 34s (95 B/s)
Reading package lists... Done
W: The repository 'https://deb.nodesource.com/node_10.x sylvia Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: GPG error: http://repository.spotify.com stable InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 4773BD5E130D1D45
W: The repository 'http://repository.spotify.com stable InRelease' is not signed.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'https://mirrors.evowise.com/mariadb/repo/10.1/ubuntu serena Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'https://deb.opera.com/opera-stable stable Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
E: Failed to fetch https://deb.nodesource.com/node_10.x/dists/sylvia/main/binary-amd64/Packages  404  Not Found
E: Failed to fetch https://mirrors.evowise.com/mariadb/repo/10.1/ubuntu/dists/serena/main/binary-amd64/Packages  404  Not Found
E: Failed to fetch https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu/dists/serena/main/binary-amd64/Packages  404  Not Found
E: Failed to fetch https://deb.opera.com/opera-stable/dists/stable/non-free/binary-amd64/Packages  503  Backend unavailable, connection timeout
E: Some index files failed to download. They have been ignored, or old ones used instead.

Sofware Manager comments from other users are things like "go here for the latest version" (https://github.com/nodesource/distributions/blob/master/README.md#manual-installation)

Sofware Manager comments from other users are things like "go here for the latest version" (https://github.com/nodesource/distributions/blob/master/README.md#installation-instructions)

What I then tried

It seemed that failing sudo apt-get update was at the root of the problem.

Inspired by other questions I tried this:

sudo rm /var/lib/apt/lists/* -vf

followed by this:

sudo apt-get update

More lists of errors and warnings that don't mean much to me:

...
Err:40 https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena/main amd64 Packages
  404  Not Found
Ign:41 https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena/main i386 Packages
Ign:42 https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena/main all Packages
Ign:43 https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena/main Translation-en_GB
Ign:44 https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena/main Translation-en
Fetched 329 kB in 28s (11.6 kB/s)
Reading package lists... Done
W: An error occurred during the signature verification. The repository is not updated and the previous index files will be used. GPG error: http://repository.spotify.com stable InRelease: The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 4773BD5E130D1D45
W: The repository 'https://mirrors.evowise.com/mariadb/repo/10.1/ubuntu serena Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: The repository 'https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu serena Release' does not have a Release file.
N: Data from such a repository can't be authenticated and is therefore potentially dangerous to use.
N: See apt-secure(8) manpage for repository creation and user configuration details.
W: Failed to fetch http://repository.spotify.com/dists/stable/InRelease  The following signatures couldn't be verified because the public key is not available: NO_PUBKEY 4773BD5E130D1D45
E: Failed to fetch https://mirrors.evowise.com/mariadb/repo/10.1/ubuntu/dists/serena/main/binary-amd64/Packages  404  Not Found
E: Failed to fetch https://mirrors.evowise.com/mariadb/repo/10.2/ubuntu/dists/serena/main/binary-amd64/Packages  404  Not Found
W: Some index files failed to download. They have been ignored, or old ones used instead.