  We study the non-equilibrium phase diagram of a fully-connected Ising
$p$-spin model, for generic $p>2$, and investigate its robustness with respect
to the inclusion of spin-wave fluctuations, resulting from a ferromagnetic,
short-range spin interaction. In particular, we investigate the dynamics of the
mean-field model after a quantum quench: we observe a new dynamical phase
transition which is either first or second order depending on the even or odd
parity of $p$, in stark contrast with its thermal counterpart which is first
order for all $p$. The dynamical phase diagram is qualitatively modified by the
fluctuations introduced by a short-range interaction which drive the system
always towards various paramagnetic phases determined by the strength of time
dependent fluctuations of the magnetization.
