I’m not sure that example is particularly interesting, because it only modifies ask. If you want to modify ask you can just use local! It seems that it is the examples that modify local that are interesting. such as your example at What is a higher-order effect? - #18 by ymdfield.
I’m not sure I understand fully what you mean, but what the handle actually is depends on the effect in question. You can see them all in the source. For example, State s is an IORef s, Reader is a State, Exception e is a forall a. e -> IO a that throws an exception, Coroutine a b is a function a -> Eff e b, etc., etc.