I wouldn't say "represents" an increase in kinetic energy. For sure, the kinetic energy is not the only form of energy in H. Searching some exotic conditions, it may be possible that U contains no kinetic energy at all.
There is more. The enthalpy of some amount of matter is not fully contained in that matter, as opposed to the internal energy: PV results from the surroundings. So, attempts to represent H by attributes of the amount of matter, like for instance the kinetic energy of atoms of that matter, will fail.
The nearest relationship is in an adiabatic expansion of gas, where enthalpy fully converts in macroscopic kinetic energy. But that's only one situation.