cPanel has an option called “Cookie IP Validation”. This setting checks that the IP address that the visitor originally came from matches their IP address on subsequent requests. With CloudFlare, the IP address for requests to your server can variably come from any IP address in the list at https://www.cloudflare.com/ips
To resolve this issue, firstly disable temporarily cloudflare, then in your WHM, navigate to
"Tweak Settings" -> "Security" -> "Cookie IP Validation". On this screen, set the value for
"Cookie IP Validation" to "disabled". After this, loading your cPanel WHM interface through CloudFlare should work as expected.