WineHQ on Ubuntu 19.10

does not have a release file indicate the repository has no support for your Ubuntu release, in this case it's Ubuntu Eoan → 19.10.

You have added deprecated/outdated repository, the developer has put up an announcement regarding this issue on the WineHQ mailing list https://www.winehq.org/pipermail/wine-devel/2017-March/117104.html.

You have to follow the new guide for installing Wine, luckily the Wine team has made this easy by posting a complete guide on their site— which you can uses with your 19.10.

sudo dpkg --add-architecture i386 
wget -nc -qO- https://dl.winehq.org/wine-builds/winehq.key | sudo apt-key add -
sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ eoan main'

Then install the packages.

sudo apt install --install-recommends winehq-staging

_{According to https://dl.winehq.org/wine-builds/ubuntu/dists/eoan/main/binary-amd64/, the only available version is: staging and devel}

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UPDATE

The new repo works fine, it's just you haven't removed old obsolete repo which you added before.

sudo add-apt-repository --remove ppa:wine/wine-builds
sudo apt update

Now there should not be any error.

My solution for 19.10; (It's worked for me on Ubuntu 19.10)

Don't use this repository:

sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ eoan main'

Use this repository:

sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/bionic main'

"eoan main" is recommented by winehq official website but not works. If you trying using 19.04 repository on 19.10, its works fine :)

If you want to use WineHQ for 19.10, enter this commands;

(Switch to "root" user.)

sudo dpkg --add-architecture i386 
wget -nc https://dl.winehq.org/wine-builds/winehq.key
sudo apt-key add winehq.key
rm -rf winehq.key
sudo apt-add-repository 'deb https://dl.winehq.org/wine-builds/ubuntu/ disco main'
sudo apt update
sudo apt install --install-recommends winehq-stable

For testing, enter this command:

winecfg