The singularity is a lie. By which I mean, it's not as if this mass is all really in one infinitesimal dot...it's just that you do any characterization of that mass when even light can't escape. So no, you wouldn't be destroying the singularity since we don't really know that's what it is, but, if they do happen to change the black hole by adding charge and angular momentum, and it allows radiation to escape it will cease to be a black hole.
I would like to find a black hole that's just barely massive enough and then try this.
Finally, I think they're just trying to do a thought experiment whereby they change the shape or topology of the event horizon. Imagine a toroidal event horizon for example.