We study magnetic properties of spin glass (SG) systems under a random field (RF), based on the suggestion that RFs can be induced by a weak transverse field in the compound LiHo x Y 1 − x F 4 .Weconsideracluster 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 Δ. We adopt the one-step replica symmetry breaking approach to get an exactly solvable single-cluster problem. We discuss the behavior of order parameters, specific heat C m , nonlinear susceptibility χ 3 , and phase diagrams for different disorder configurations. In the absence of RF, the χ 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 Δ. However,the T f is displaced to lower temperatures, enhancing considerably the ratio T** /T f . Furthermore, the divergence in χ 3 is replaced by a rounded maximum at a temperature T*, which becomes increasingly higher than T f as Δ is enhanced. 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 χ 3 behavior triggered by RF and the deviation of the conventional relationship between the T f and T**
