Customizing REST API Gateway Authorizer Response

Yes, you can return a specific response from Lambda Authorizer, you have access to a special object called $context.authorizer which you can use to customize the response, please review this section of the documentation which explains in depth, you can use in your response template.

In the API Gateway console you can change the response to custom for Access Denied (403), Under the Response templates -> Template body you can specify the specific field being returned from your authorizer, In my case, I have a authorizer for my API where I return a message field populated with a custom message under the context key for denied requests, so my Template body is {"message":$context.authorizer.context.message}

I have this field populated with this specific string "context": {... "message": "You do not have access to this resource"} so when authorizer denies the request, the response I get is {"message":You do not have access to this resource}

Please see this SO post as well that explains it.

HTH