May a star explosion be in that way weak that the expelled material gradually recollapse and form the star again? May a star explosion be in that way weak that the expelled material gradually recollapse and form the star again? My question is not about would that explosion leave a neutron star as a remnant but just if the gravity of particles involved could force a formation of a new version of the old star from the same material.

 A: When will something that explodes in space fall back? If we start with a spherical mass it will have a binding energy $\sim 3GM^2/5R$. That is the energy difference between the current state and all particles escaping to infinity. So any explosion with less energy that that will produce fallback... in principle, but not in practice.
The problem is that energy gets divided unevenly among the component particles. A plausible first guess is that their speeds $C$ will follow the Maxwell-Boltzman distribution $$\Pr[c<C<c+dc]=4\pi c^2 \left(\frac{m}{2\pi k_B T}\right)^{3/2}e^{-\frac{mc^2}{2k_B T}}.$$ That distribution has a long tail of high-velocity particles that may exceed the escape velocity of the star, $\sqrt{2GM/r}$. For big, cold objects the rate of escape tends to be small. But if you blow up a star, even if with less energy than its binding energy, it will get pretty hot!
If we first magically distribute an explosion energy corresponding to the binding energy equally among particles (it quickly thermalizes to the above distribution) it would be $(3GM^2/5R)(m/M)$ per particle, or an equivalent temperature $T\sim 3GM m/5k_B R $. For the sun, this is about 13 million degrees. In this case the most probable speed is 478 km/s, which is 77% of escape velocity. But the mean speed is $\sqrt{4/\pi}$ times that, 86% of escape velocity.
Hence, we should expect a lot of gas (likely more than a third) to just escape even if it was an explosion slightly less than the entire binding energy. For sufficiently small explosions less escapes, but at this point it may look more like a brief expansion rather than a proper explosion.
A supernova is about $10^{46}$ J, 100,000 times the solar binding energy. Even a 100 solar mass star will be disrupted by that.
So for a big enough star that explodes in a supernova, there may definitely be enough material left over to form a new, much smaller star. But I suspect usually the hot environment (heated by radiation from accretion of the gas onto a neutron star or a black hole) tends to heat up and then dissipate the gas. It merges with the interstellar medium near the supernova shock front and eventually cools down enough to start condensing into stars elsewhere, without staying gravitationally bound.
A: In the mathematical models that astrophysicists have developed to describe the different ways that stars end their lives, there exists the possibility that the shock wave produced by core collapse inside a big star can get stalled on its way out and not make it all the way to the surface, or that gravity might be strong enough to pull the ejecta of an exploding star back before it can escape.
Stellar collapse modeling is an extremely complicated task, about which much has been written (too much to summarize here). A good book for nonspecialists on this topic is Cosmic Catastrophes by J. Craig Wheeler.
