I've been frustrated by this too so I submitted a support request to Mapmyfitness and this was the unsatisfactory reply. I'm sure that I'm not the first person to complain about this.
Hi Don,
Thanks for writing. I apologize for any inconvenience this has caused. Cycling workouts do sync to Apple Health but they are currently syncing as general workotus rather than specifically as cycling workouts so you will not see them when attempting to view the cycling distance. Please know that I have forwarded this along to the development team and they are investigating the cause. At this time, I'm unsure of how quickly it will be fixed, but we do hope to have the problem taken care of soon. We will to try our best to reach out to you again when a fix for this issue is available. Thank you for your patience and understanding while we troubleshoot the problem.
If you have any additional information that may aid in our investigation, please let me know.
Kindly,
Tim