No, it cannot. A black hole is something vastly different from a star. It's vastly different from anything else. It's not a thing, really, but more like a portion of very distorted spacetime. Nothing escapes from it simply because there is no way out - spacetime is distorted in such a way that all trajectories lead to the center.
Now, there is a mechanism where radiation is generated just outside the black hole, sucking energy from its gravitational field and therefore from its mass. This is called Hawking radiation. It does not come from inside the BH, but the way it's created it leeches the BH's energy / mass. In time, this would diminish the mass of the BH.
It turns out, the smaller the BH, the stronger the Hawking radiation. This in turn makes the BH lose weight even faster, so the HR gets even stronger, and so on. It's a vicious cycle. If the black hole is small enough, the process turns into an explosion - HR gets so intense, the whole mass of the black hole is converted into radiation at once. It's not a supernova, not as big as that, but it is a powerful explosion.
The only issue here is - HR is very weak when BHs are big. So regular BHs will take an incredibly long time to slim down, if ever, to the tiny size where HR spirals out of control. Basically this never happens in reality.
TLDR: Not a supernova, just a huge bomb, but not very likely.