I got this error: "Child failed to make LIVEAPI connection to cPanel." in cpanel account when i clicked to access Cloudflare section.
That WHM server is using cpanel cloudflare plugin.

I tried to run command:

curl -vs https://api.cloudflare.com/host-gw.html -d "act=cpanel_info" -d "host_key=MYCloudflareHostKeyHERE"
not helped

still nothing.

Checking version of the cloudflare cpanel plugin:

# grep version /usr/local/cpanel/base/frontend/paper_lantern/cloudflare/config.js
"version": "6.1.5"

And this worked - update:

# bash <(curl -s https://raw.githubusercontent.com/cloudflare/CloudFlare-CPanel/master/cloudflare.install.sh) -k MYCloudflareHostKeyHERE-n 'Company Name Here'
Starting CloudFlare CPanel Installation...
Downloading and unpacking latest version v7.0.1...
info [install_plugin] Installing to paper_lantern
info [install_plugin] Running sprite_generator
info [install_plugin] Queueing task to update API spec files
Plugin installed ok
Cleaning up