I had the same problem, you must setup a Page Rule in cloudflare.
Example:
Pattern:
http://mysite.com/admincp/*
And as for the Page Rule options, Disable CloudFlare’s caching and optional performance features for that Page Rule.
Also create another Page Rule for "http://mysite.com/modcp/*" and do the same with the Page Rule options as the one above.
Note: If you renamed your AdminCP and/or ModCP folders, you will need to use the correct folders. Also note the "*" (wildcard) at the and of the URL for the Patterns. Them Page Rules will match the entire contents of the directory using the Wildcard. That should stop the AdminCP and ModCP from always refreshing.
I would setup the 2 page rules, so you can still use cloudflare with your site, instead of disabling cloudflare globally.