One of sites is currently hosted on a server using HSpehere control panel and I am planning to move the site to a new host offering cpanel.
I think the in case of cPanel, the backup file import process checks for specific cPanel details when importing the backup file.
What is the easiest way to migrate the site for my new host, if it is possible?
The easy way to avoid the issue would be , take the backup of your webfiles and databases. once you got your account created on cpanel server , restore the backup ( Non cpanel backup ) and restore the DB ( .sql format ) . This would be the easiest way . It is lengthy but it will avoid the issues.
If you've got access to the CP server all the info is in there in the HSphere PGSQL database. We're part way through some scripts to automate the data extraction, if I get round to it we'll be trying to assemble a cPanel backup file from the data as well, as we've got a number of accounts to move still.