I wanted to do the same thing, and eventually accomplished it by using the Mockito “spy” function, which wraps a real object with a mock object. All calls to the mock object get forwarded to the real object, except those you are trying to mock. For example:
ObjectMapper om = Mockito.spy(new ObjectMapper());
Mockito.when( om.writeValueAsString(ErrorObject.class)).thenThrow(new JsonProcessingException("") {});
All usages of om will be handled by the underlying ObjectMapper instance until an instance of ErrorObject gets passed in, at which point the JsonProcessingException will be thrown.
The newJsonProcessingException is created as an anonymous class, as it is a protected class and only a sub-class can be instantiated.