Greetings,
Thank you for asking.
I remember this comes up on topics lately much more than ever before and I am sorry to hear that you’re experiencing this 
Before moving to Cloudflare, was your Website working over HTTPS connection?
May I ask what was the error code? 403 or 520 or 521 shown in the Devloper console (F12) of your Web browser for the particular admin-ajax.php request? 
I wonder if any of Cloudflare security & protection settings like Bot Fight Mode or Browser Integrity Check challenged or blocked the request
Kindly you check and provide some more details about which service did it trigger and got that result in the Cloudflare dashboard → Security → Overview for the past 24hours or so. Once you find them, click on a particular one to find more details about it (user-agent, IP, HTTP version …).
Just in case, related to the WordPress, I’d suggest you to whitelist your origin host / server / hosting IP address by navigating to the Security → WAF → Tools → IP Access Rules with the action “allow” for your Website and try again.
It knows to happen due to the WordPress using HTTP/1.0 and empty user-agent, therefore while executing WP-Cron or some other related JSON/REST API request via plugin.