When Health Sync tells you that there is an HMS Core error, then you need to reset the HMS Core app on your phone. Hopefully the issue with the HMS Core error will be solved soon. It has been reported to Huawei.
For now the solution is as follows:
First go the list of apps in the “Apps” Settings menu. First open the Settings menu, then Apps, and on some phones, one more time you need to select “Apps”.
In the ‘Search apps’ field, write: HMS Core
Select the HMS Core app
select the Storage menu
Select MANAGE SPACE
Now click on CLEAR DATA. This doesn’t do any harm. But you will have to authorize Health Sync again for Huawei Health. Health Sync will show you what to do when you open the app. Maybe you will get some other service that asks for authorization. This depends on the apps you use on the phone.