diff --git a/moto/s3/exceptions.py b/moto/s3/exceptions.py
index 39047bf44..1587d85b4 100644
--- a/moto/s3/exceptions.py
+++ b/moto/s3/exceptions.py
@@ -5,7 +5,7 @@ ERROR_WITH_BUCKET_NAME = """{% extends 'single_error' %}
 """
 
 ERROR_WITH_KEY_NAME = """{% extends 'single_error' %}
-{% block extra %}<KeyName>{{ key_name }}</KeyName>{% endblock %}
+{% block extra %}<Key>{{ key }}</Key>{% endblock %}
 """
 
 ERROR_WITH_ARGUMENT = """{% extends 'single_error' %}
@@ -91,9 +91,11 @@ class MissingBucket(BucketError):
 class MissingKey(S3ClientError):
     code = 404
 
-    def __init__(self, key_name):
+    def __init__(self, **kwargs):
+        kwargs.setdefault("template", "key_error")
+        self.templates["key_error"] = ERROR_WITH_KEY_NAME
         super(MissingKey, self).__init__(
-            "NoSuchKey", "The specified key does not exist.", Key=key_name
+            "NoSuchKey", "The specified key does not exist.", **kwargs
         )
 
 
diff --git a/moto/s3/models.py b/moto/s3/models.py
index 44aa2493f..8cf875ae9 100644
--- a/moto/s3/models.py
+++ b/moto/s3/models.py
@@ -1620,7 +1620,7 @@ class S3Backend(BaseBackend):
         if key is not None:
             key.set_acl(acl)
         else:
-            raise MissingKey(key_name)
+            raise MissingKey(key=key_name)
 
     def put_object_legal_hold(
         self, bucket_name, key_name, version_id, legal_hold_status
@@ -1686,7 +1686,7 @@ class S3Backend(BaseBackend):
 
     def set_key_tags(self, key, tags, key_name=None):
         if key is None:
-            raise MissingKey(key_name)
+            raise MissingKey(key=key_name)
         boto_tags_dict = self.tagger.convert_dict_to_tags_input(tags)
         errmsg = self.tagger.validate_tags(boto_tags_dict)
         if errmsg:
diff --git a/moto/s3/responses.py b/moto/s3/responses.py
index aaa0427ce..6488a93cb 100644
--- a/moto/s3/responses.py
+++ b/moto/s3/responses.py
@@ -1357,7 +1357,7 @@ class ResponseObject(_TemplateEnvironmentMixin, ActionAuthenticatorMixin):
 
         key = self.backend.get_object(bucket_name, key_name, version_id=version_id)
         if key is None and version_id is None:
-            raise MissingKey(key_name)
+            raise MissingKey(key=key_name)
         elif key is None:
             raise MissingVersion()
 
@@ -1552,7 +1552,7 @@ class ResponseObject(_TemplateEnvironmentMixin, ActionAuthenticatorMixin):
                     src_version_id=src_version_id,
                 )
             else:
-                return 404, response_headers, ""
+                raise MissingKey(key=src_key)
 
             new_key = self.backend.get_object(bucket_name, key_name)
             mdirective = request.headers.get("x-amz-metadata-directive")
diff --git a/tests/test_s3/test_s3.py b/tests/test_s3/test_s3.py
index c75966ed1..a355b706b 100644
--- a/tests/test_s3/test_s3.py
+++ b/tests/test_s3/test_s3.py
@@ -3319,6 +3319,25 @@ def test_boto3_head_object_with_versioning():
     old_head_object["VersionId"].should_not.equal(head_object["VersionId"])
 
 
+@mock_s3
+def test_boto3_copy_non_existing_file():
+    s3 = boto3.resource("s3")
+    src = "srcbucket"
+    target = "target"
+    s3.create_bucket(Bucket=src)
+    s3.create_bucket(Bucket=target)
+
+    s3_client = boto3.client("s3")
+    with pytest.raises(ClientError) as exc:
+        s3_client.copy_object(
+            Bucket=target, CopySource={"Bucket": src, "Key": "foofoofoo"}, Key="newkey"
+        )
+    err = exc.value.response["Error"]
+    err["Code"].should.equal("NoSuchKey")
+    err["Message"].should.equal("The specified key does not exist.")
+    err["Key"].should.equal("foofoofoo")
+
+
 @mock_s3
 def test_boto3_copy_object_with_versioning():
     client = boto3.client("s3", region_name=DEFAULT_REGION_NAME)
@@ -3359,7 +3378,7 @@ def test_boto3_copy_object_with_versioning():
             Bucket="blah",
             Key="test5",
         )
-    e.value.response["Error"]["Code"].should.equal("404")
+    e.value.response["Error"]["Code"].should.equal("NoSuchKey")
 
     response = client.create_multipart_upload(Bucket="blah", Key="test4")
     upload_id = response["UploadId"]
@@ -4701,6 +4720,7 @@ def test_boto3_put_object_tagging():
         {
             "Code": "NoSuchKey",
             "Message": "The specified key does not exist.",
+            "Key": "key-with-tags",
             "RequestID": "7a62c49f-347e-4fc4-9331-6e8eEXAMPLE",
         }
     )
@@ -4760,6 +4780,7 @@ def test_boto3_put_object_tagging_on_earliest_version():
         {
             "Code": "NoSuchKey",
             "Message": "The specified key does not exist.",
+            "Key": "key-with-tags",
             "RequestID": "7a62c49f-347e-4fc4-9331-6e8eEXAMPLE",
         }
     )
@@ -4828,6 +4849,7 @@ def test_boto3_put_object_tagging_on_both_version():
         {
             "Code": "NoSuchKey",
             "Message": "The specified key does not exist.",
+            "Key": "key-with-tags",
             "RequestID": "7a62c49f-347e-4fc4-9331-6e8eEXAMPLE",
         }
     )