Enthalpy H and internal energy U are both state functions
Irrespective of the pressure and volume changes from initial to final state, you can consider H or U to describe processes.
Enthalpy H is most useful for processes at constant pressure
If you are interested in the amount of heat a reaction gives off, you have to consider that a closed system can exchange energy with the surrounding via heat or work. In the special case of a reaction (or process) where the only work is P-V-work, and where the pressure is constant, the heat transferred is equal to the change in enthalpy. That is why reactions with negative change in enthalpy are called exothermic. If their is non-P-V work, however, an exothermic reaction can happen without giving off heat (e.g. reaction in a battery).
Internal energy is closely related to heat when the volume is constant
In the special case of a reaction (or process) where there is no work, the heat transferred is equal to the change in internal energy. This happens, for example, when the volume is constant and there is no non-P-V work (a constant volume implies that there is no P-V-work).
What about enthalpy changes when the pressure does change?
In that case, there is no easy relationship between heat and enthalpy. At constant volume, VΔP is the difference between enthalpy and internal energy. There is no P-V-work, and it makes sense to work the the internal energy U instead.
In this case, what is VΔP?
It is the difference between H and U in the case of constant volume, like stated in the question. It does not have any physical meaning. The way H is defined makes it easy to interpret for constant pressure processes. The flip-side is that it is difficult to interpret for all other processes, including those happening at constant volume.
Maybe a form of energy transfer that is not work but similar to it? Does it have particular name? Or can it be included in a kind of works?
No, there is no work associated with it. It is just what is mathematical left over from the difference between U and H when the volume is constant.