On Esko Cloud: Your Automation equipment is member of the group Automation Engines.
The group Automation Engines has Manager access to the site mentioned in the JSON string (above 'SomeSiteName'). You can check these permissions by going as Administrator to that site and on the root, click the Share button/Manage Permissions. You will see a line containing Automation Engines with Manager Role.
After correcting the latter two, your Automation Engine needs to log out and log in again. There is no user interface for this. One solution is to wait for an hour and try again.
If that is not successful:
Check the log file on Automation Engine. You find it on C:\Esko\bg_data_fastserver_v100\LOGGING\bgmd_dateinfo.log (where C:\Esko is your Esko home, possibly on another drive and dateinfo is different every day, but only 1 fitting today). Search for ERROR CM_TSitesHandler. If you find it, it will typically show also something like Getting userinfo(via token) failed for us.esko.cloud-xxx. HTTP-GET to https://iam.us.esko.cloud/rest/iam/getCurrentUserInfo?org=a-p-xxxx-xxxx-xxxx-xxxx returned: 400→ means that your token is wrong or no longer valid (e.g. because a newer one was meanwhile generated for the same equipment).
If you don't find the previous, search for "Connection Setup failed". A typical line you find is: Problem connecting Automation Engine Enterprise: there is configuration (a-p-xxxx-xxxx-xxxx-xxxx/SomeSiteName in us.esko.cloud) but no equipment having appropriate access rights. This line points to missing access rights (see above for what you need) but seems to also point to any other issue found.
Token does not correspond to an Automation Engine Equipment node. Check whether you are not using the token of another equipment type such as WebCenter or other. Create an equipment of type Automation Engine instead.
If none of this works, generate a new token in Esko Cloud.
If that is still not successful, you could delete your Automation Engine equipment and generate it again. However, please note that a newly created one, even if it has the same name, is a new identity in Esko Cloud. This means that files created by the previous connection will no longer be recognized as this Automation Engine. For Share & Approve, this has largely aesthetic disadvantages, but in Automation Engine 'Enterprise', this can have more far-going disadvantages.