  We study magnetic properties of spin glass SG systems under a random field
(RF), beased on the suggestion that RFs can be induced by a weak transverse
field in the compound LiHo$_x$Y$_{1-x}$F$_4$. We consider a cluster spin model
that allows long-range disordered interactions among clusters and short-range
interactions inside the clusters, besides a local RF for each spin following a
Gaussian distribution with standard deviation $\Delta$. We adopt the one-step
replica symmetry breaking (RSB) approach to get an exactly solvable
single-cluster problem. We discuss the behavior of order parameters, specific
heat $C_{m}$, nonlinear susceptibility $\chi_3$ and phase diagrams for
different disorder configurations. In the absence of RF, the $\chi_3$ exhibits
a divergence at $T_f$, while the $C_{m}$ shows a broad maximum at a temperature
$T^{**}$ around 30$\%$ above $T_f$, as expected for conventional SG systems.
The presence of RF changes this scenario. The $C_{m}$ still shows the maximum
at $T^{**}$ that is weakly dependent on $\Delta$. However, the $T_f$ is
displaced to lower temperatures, enhancing considerable the ration
$T^{**}/T_f$. Furthermore, the divergence in $\chi_3$ is replaced by a rounded
maximum at a temperature $T^{*}$, which becomes increasingly higher than $T_f$
as $\Delta$ enhances. As a consequence, the paramagnetic phase is unfolded in
three regions: (i) a conventional paramagnetism ($T>T^{**}$; (ii) a region with
formation of short-range order with frozen spins ($T^{*}<T<T^{**}$); (iii) a
region with slow growth of free-energy barriers slowing down the spin dynamics
before the SG transition ($T_f<T<T^{*}$) suggesting an intermediate Griffiths
phase before the SG state. Our results reproduce qualitatively some findings of
LiHo$_x$Y$_{1-x}$F$_4$ as the rounded maximum of $\chi_3$ behavior triggered by
RF.
