That $\mathcal H'_g$ can't be connected for $g \geq 3$, isn't this just the "lift to Teichmuller space" of the result that the mapping class group isn't the hyperelliptic group?
In particular, the path components of $\mathcal H'_g$ are indexed by the cosets of the hyperelliptic group in the mapping class group. No?
edit: Anweshi, I'm not sure how you're thinking about Teichmuller space but the answer to your question can be seen in many ways, you don't have to use the language of orbifolds, it's just a convienient container. In my mind I suppose I think of a path in Teichmuller space as a motion of the surface -- make this concrete using Fenchel-Nielsen coordinates, for example. So if you have a path that connects one point to another there is an associated diffeomorphism of the surfaces that stretches/twists the metric appropriately and matches up the markings of the surfaces. So if you go between two points in your $\mathcal H_g'$ covering the same point in $\mathcal H_g$ the relating diffeomorphism is in the hyperelliptic group (since the hyperelliptic group is a subgroup of the mapping class group). This is how you `see' the cosets of the hyperelliptic group in the mapping class group as indexing $\pi_0 \mathcal H'_g$.