Maybe. Although, "cyclicality" as such may not make sense at all, because in extreme conditions like the big bang, some kind of still incomprehensible garbage occurs over time. Either it shrinks to a point, or it wraps itself up in some cunning makar.
As for black holes - no, they have very little to do with the big bang, if only because they evaporate gradually, very slowly. They definitely cannot "explode", and even more so, give birth to a new universe.
On the topic of new universes emerging "from the ashes" of old ones, you can google the "bubble universe" - "bubble universe"
I'm not an expert in this topic, but I've heard about two possible options for the development of the fate of our universe. In simple terms, some stars form supernovae that explode relatively quickly to form a neutron star. If at the same time the neutron star itself is so massive that the pressure of the neutron gas cannot withstand gravitational compression, then a collapse will occur, namely the formation of a black hole. Dying stars also become black holes. One way or another, our universe is filled with black holes that absorb energy, growing in mass.
And here two scenarios are possible.
Nothing prevents one black hole from swallowing another, if they are close enough to each other. But here the fact interferes with the fact that our universe is rapidly expanding. If the expansion "wins", then the black holes will be too far apart, and without the influx of new mass, they will begin to lose energy due to photons tunneling through the event horizon. Stars will go out and black holes will start to evaporate. Thermal death of the universe will occur.
However, if black holes "win", then, absorbing each other, sooner or later they will reach the limit when the force of gravity becomes so great that it will compress all the available mass into a singularity ... Here's a new big bang and a new birth of the universe from the energy of the old.