Alrighty, there are some fundamental flaws in your model. I'm by no means a super-expert on black hole theory, or conversely, white hole theory, but I'll try to explain the flaws as best as I could.
First off, matter and light entering a black hole is not "destroyed". You had the idea right until you used that word, basically, yes, it is compressed into the singularity, ie: if you could imagine that it is crushed from all directions until it is squeezed into an impossibly small dot, with no height, width or depth. Essentially, all the volume of the matter falling into the black hole is compressed into a zero-dimensional point, however, the mass remains, and increases the total mass of the black hole, and due to this, increases the gravitational pull of the black hole.
An extreme example of this, is let's assume that out of some cosmic fluke, a black hole that has one solar mass randomly (and yes, impossibly) forms within orbit around our sun. The sun and the black hole would have the exact same mass, and would attract each other, dancing in circles around each other until they were close enough that solar material was pulled into the black hole. As the material fell into the black hole, the physical size of the black hole would not increase, however, it's mass would, until it had completely consumed the sun, at which point it would have doubled in mass. This would increase it's gravitational pull, but the fact of the matter is that physically, it's still that zero-dimensional dot.
Now, as far as white holes, they are extremely intriguing, and physicists are already arguing about whether or not one may have been observed. Here is my theory regarding their possible existence. Black holes can last for millenia, white holes, if they can exist, probably only last for nanoseconds to minutes (as was the "potential white hole" gamma ray burst observed in 2006, which lasted for 102 seconds). We already know the black holes can collapse and destabilize. If this happens, and for whatever reason the gravitational field starts falling apart, could this not result in an explosion of the singularity, dumping all that condensed matter and information back into space? This could be a white hole, and while not necessarily the long sustained effect we see in a black hole, is quite possibly a solution to this problem without really interfering with the second law of thermodynamics... any thoughts?