  We study the stochastic relaxation dynamics of the Ising p-spin model on a
random graph, a well-known model with glassy dynamics at low temperatures. We
introduce and discuss a new closure scheme for the master equation governing
the continuous-time relaxation of the system, that translates into a set of
differential equations for the evolution of local probabilities. The solution
to these dynamical mean-field equations describes very well the
out-of-equilibrium dynamics at high temperatures, notwithstanding the key
observation that the off-equilibrium probability measure contains higher-order
interaction terms, not present in the equilibrium measure. In the
low-temperature regime, the solution to the dynamical mean-field equations
shows the correct two-step relaxation (a typical feature of the glassy
dynamics), but with a relaxation timescale too short. We propose a solution to
this problem by identifying the range of energies where entropic barriers play
a key role and defining a renormalized microscopic timescale for the dynamical
mean-field solution. The final result perfectly matches the complex
out-of-equilibrium dynamics computed through extensive Monte Carlo simulations.
