If it's a hosted feature service (published to the ArcGIS Online cloud), you'll have to edit the supported operations in the item description of the service when logged into ArcGIS Online. You can always check which operations are allowed by checking the REST endpoint of the service, scroll to the bottom and see what is listed under supported operations.
You will have to see the feature server for the layer to see the operations like create, delete etc. The rest endpoint link posted is for the mapserver which doesn't have those operations on it. Take a look at the sample services for reference
Supported Operations: Query Apply Edits Add Features Update Features Delete Features Generate Renderer Return Updates
But 1. When I add or edit features the data is not saved. 2. When editing newly added data editor window closes and disappears, data not saved. When editing data existing in the geodatabase - changes are not saved and the window closes. OK button - becomes inactive, the Delete button - active What could be wrong? PS. Delete operation is performed normally (in geodatabase too). But if I view my feature layer in ArcGIS.com Map - all works normally, edited changes are saved in my geodatabase.
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