Cloudflare could cause drop in Google SERPS positions if there is incorrect host configuration. Technically the traffic passes through the proxy (filtered by several layers and firewall rules). Infact cloudflare can influence rankings. For many small sites you dont even need cloudflare. Google has said speed is not everything.
A decade ago, a web connection is made only between a server a client direct. Today cloudflare acts as third party which could slow down traffic. Besides cloudflare hosts hundreds of sites on same plan and there is no possibility to get a dedicated private ip under free plan.
I have been investigating this problem for a while..
Investigate these issues if you see a huge drop in google organic rankings.
SSL issues
Incorrectly configured SSL can cause issues. Fix mixed content. You must install valid SSL cert (eg. Lets encrypt) in your local server, else it wont work.
Robots.txt
Check this file for causing any blocks preventing google to crawl your site.
Rocket Loader
Know how have issues with SEO. Better off you disable.
Crawler Hints
May cause issues blocking off unknown crawlers.
Bot fight mode
Can block bots from unknown search engines and bots preventing you go get good backlinks.
Hot Link protection
Could prevent your results appearing in Google discover
Firewall
Check your WAF firewall rules that could prevent certain IPs or sites.
