May I ask is the subdomain like sub.yourdomain.com or www.sub.yourdomain.com in terms of an possible “subdomain deep-level issue”?
Before moving to Cloudflare, was your Website working over HTTPS connection?
If so, did you had an valid SSL certificate installed at your origin host / server which covers both your naked (root) domain any any other needed sub-domain like www, mail, etc.?
May I ask what SSL option have you got selected under the SSL/TLS tab at Cloudflare dashboard for your domain ( Flexible, Full, Full Strict … )?
In case you do not have an SSL certificate, you can use Cloudflare SSL, if so, kindly make sure you follow the instructions as follows on the below article to setup an SSL certificate using Cloudflare Origin CA Certificate: