Could a nuclear reactor run (controllably) in Prompt-Critical mode? It would have a much higher power output than a normal nuclear reactor. Could it be controlled by some high-tech material that absorbs neutrons when an electric current is applied to it (so the reaction time would be fast enough)?
This question doesn't make much sense in terms of pure terminology.
A prompt neutron is a neutron which is emitted right when the fission event happens. For a recap, a fissile nucleus absorbs a neutron and then splits into two large fragments and makes several smaller particles in the process, which usually includes 2 or 3 free neutrons and a neutrino. A delayed neutron is emitted spontaneously by one of those large fragments after the initial fission.
You could say that all free neutrons in a nuclear reactor are either prompt or delayed. The fact that you have delayed neutrons (ANY delayed neutrons) would seem to indicate that any prompt critical state will have k>1, meaning that power is increasing. You can have k>1 without being prompt critical (this is often desirable), but you can not have k=1 when the reactor is prompt critical. Also, the fraction of neutrons which are delayed doesn't change very much between different fuel types. Even fast reactors have delayed neutrons.
Generally you operate a reactor at k=1, and this is absolutely necessary for any stable operation. This prompt critical state only applies for transients. So by talking about "running" at prompt critical state, the question is trying to fit a square peg in a round hole.
So my answer would basically be "no". If it's prompt critical, then it is necessarily increasing in power and you can't "run" a reactor like that.
The point of prompt criticality is there are enough prompt neutrons to keep the reaction going. Prompt neutrons are produced something like 0.001 nanoseconds (1E-12 seconds) after the neutron is absorbed.
If you want to control something that is prompt critical, you need a control system that can react in that type of time frame. Even some hypothetical electronic control rods would not be able to keep up with that. Transistors and the speed of electrical propagation (aka the speed of light) are just not fast enough.
I remember my professor telling me "If it were not for delayed neutrons, all we could make are bombs."
I know of no credible mechanism that can create a super prompt critical situation in a nuclear reactor; delayed neutrons determine the behavior. Without delayed neutrons a reactor would be almost impossible to control. Regardless, a reactor cannot explode like a nuclear weapon. For a thermal reactor, the neutron generation lifetime is too long. For a fast reactor (and a thermal reactor) there is no mechanism for creating and maintaining a super prompt critical assembly sufficiently long for significant release of energy from fission. You have to really work hard to assemble the correct material to create a nuclear weapon.