diff --git a/swh/objstorage/cloud/objstorage_azure.py b/swh/objstorage/cloud/objstorage_azure.py
--- a/swh/objstorage/cloud/objstorage_azure.py
+++ b/swh/objstorage/cloud/objstorage_azure.py
@@ -101,7 +101,7 @@
                 container_name=self.container_name,
                 blob_name=hex_obj_id)
         except AzureMissingResourceHttpError:
-            raise ObjNotFoundError('Content %s not found!' % hex_obj_id)
+            raise ObjNotFoundError(obj_id)
 
         return gzip.decompress(blob.content)
 
diff --git a/swh/objstorage/cloud/objstorage_cloud.py b/swh/objstorage/cloud/objstorage_cloud.py
--- a/swh/objstorage/cloud/objstorage_cloud.py
+++ b/swh/objstorage/cloud/objstorage_cloud.py
@@ -139,7 +139,7 @@
         try:
             return self.driver.get_object(self.container_name, hex_obj_id)
         except ObjectDoesNotExistError as e:
-            raise ObjNotFoundError(e.object_name)
+            raise ObjNotFoundError(obj_id)
 
     def _put_object(self, content, obj_id):
         """Create an object in the cloud storage.
diff --git a/swh/objstorage/tests/objstorage_testing.py b/swh/objstorage/tests/objstorage_testing.py
--- a/swh/objstorage/tests/objstorage_testing.py
+++ b/swh/objstorage/tests/objstorage_testing.py
@@ -90,9 +90,11 @@
     @istest
     def get_missing(self):
         content, obj_id = self.hash_content(b'get_missing')
-        with self.assertRaises(exc.Error):
+        with self.assertRaises(exc.ObjNotFoundError) as e:
             self.storage.get(obj_id)
 
+        self.assertIn(obj_id, e.exception.args)
+
     @istest
     def check_missing(self):
         content, obj_id = self.hash_content(b'check_missing')