The data source is a feature service published to a federated ArcGIS Server Site as both a map service and a feature service.
An Item was created in AGO (not the Portal federated with the server) by using the feature service REST point URL.
The service item created previously was added to the web map in AGO by adding it from "My Content" (rather than adding the REST URL directly).
I tried adding a refresh rate to the properties of the table within the web map by electing to ignore the refresh rate stored with the service. After electing this change, only some of the properties were stored with the web map and the others with the service, one of which is the refresh rate.
I checked server manager to find out if I could change the refresh interval for the service, but was not successful in finding a parameter to control this.