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
Note: Starting with version 64 of WHM this is no longer an issue. Contact your hosting provider or cPanel support if you still see issues.
The solution is to access the cPanel server on a URL that doesn't proxy through Cloudflare. Please use cPanel's proxy subdomain URL http://cpanel.domain.com and have the DNS point directly to the cPanel server which is a default DNS setting.
Make sure this subdomain is not proxied through Cloudflare (not orange clouded).