Glancing at your WPT report, a good starting point will be for you to run mysqltuner.
Then implement all mysqltuner suggested tunings.
Also, check to make sure you’re using the InnoDB storage engine for all tables, as MyISAM has so many problems there’s been talk about removing MyISAM altogether.
I know I’ll likely catch flack for saying this…
You’re using CloudFlare… the source of many problems… all outside your control… which you can never fix…
Also, day to day, sometimes CloudFlare works + sometimes fails either completely or worse… intermittently… which is another source of impossible to debug problems…
First step of dealing with speed issues, for me, is always remove all cruft code between Apache + visitors.
So right now, you’re trying to debug likely CloudFlare injected problems, with no control over CloudFlare.
First step… Remove CloudFlare, which is sufficient to fix many site problems.