In order for WHM to allow sending emails from a specific Cpanel user and WordPress via plugins like EasySMTP, one needs to add that particular user to the CSF (firewall) configuration file of the server:
/etc/csf/csf.conf
in a comma-separated list like this:
SMTP_ALLOWUSER = "cpanel,user1,user2"
and then restart CSF from the console:
csf -r
This needs to be done by the server administrator!