Thanks Draka.....
I think we might be talking at cross purposes.
In Cpanel there is a cloudflare option - [ at least on shared hosting - I don't think/know it is available in the reseller... ]- from which you can sign up for and administer your cloudflare account/settings. It gives you the option of setting up one of two ways. see
https://support.cloudflare.com/hc/en-us/articles/203685674-Full-setup-versus-Partial-CNAME-setup
As far as I know - which is not a lot - you don't REQUIRE a cloudflare plug-in for wordpress - although perhaps it optimises it further or is an alternative to the cpanel option.
I suspect you are using the full DNS option . For the other, the cname would have needed to be modified with your authority:- ie whilst logged into cpanel or manually by you . Did you have to point you nameservers with your registrar to cloudflare?
I think you've answered my question though. I will probably move to the full DNS option after I open a few more services with HawkHost
Cheers