E: Package 'mysql-client' has no installation candidate in php-fpm image build using docker compose
php:7.3-fpm now use Debian 10 (Buster) as its base image and Buster ships with MariaDB, so just replace mysql-client with mariadb-client should fix it.
If you still want to use the mysql client, it's called default-mysql-client now.
I had the same issue in php:7.1-apache. Replacing mysql-client with mariadb-client works for me. Thank you.