If you are getting the following error message:

PDO::__construct(): The server requested authentication method unknown to the client [mysql_old_password]

You will need to open a support ticket with your web hosting provider.

The problem is your webhost is using a new version of MySQL client library that expects to connect with MySQLs newer 41-byte password format but the MySQL user on your MySQL server is using an old 16-byte password hash. This is causing the database connection to fail resulting in SQL errors. You will need to have your web hosting provider update old password hashes to new.

1) Run this command:

UPDATE mysql.user SET Password=PASSWORD('newpass')
WHERE User='root' AND Host='localhost';

Importance Note
Swap newpass, root and localhost in the query above with your actual MySQL username, password and hostname found in your API/database.php file.

The server admin for your web hosting provider should definitely be able to handle this task for you and when its complete, this will resolve the database connection errors.