We study the magnetic specific heat Cm and the non-linear susceptibility &chi3 following the suggestion that a random field (RF) can be induced by a transverse field Ht. This mechanism was proposed to explain how the characteristic sharp peak in the &chi3 marking a Spin Glass (SG) second order phase transition at the freezing temperature Tf in the Ising compound LiHoxY1-xF4 is replaced by a rounded maximum when Ht is applied. We analyse this problem by adopting a cluster version for the Sherrington-Kirkpatrick model in the presence of RFs [S.M. A. Tabei et al, Phys. Rev. Lett. 97, 237203 (2006)]. The clusters are (see M. V. Romitti et al, Phys. Rev. B 99, 014203 (2019)) used to improve previous mean-field treatment [C. V. Morais Junior et al, Phys. Rev. B 93, 224206 (2016); S. G. Magalhaes et al, Phys. Rev. B 95, 064201 (2017)] allowing to include long-range disordered interactions among clusters, and short-range ferromagnetic interactions within the clusters, besides a local RF. In the present study, we assume that the RF follows a Gaussian distribution with standard deviation &Delta. We adopted the replica formalism within one-step replica symmetry breaking (RSB) to get an exactly solvable single cluster problem. We analyse the behavior of Cm, &chi3, phase diagrams and the replicon for different disorder configurations and Ht. In the absence of RF triggered by Ht, the &chi3 exhibits a divergence at Tf, identifying the SG phase transition that occurs with RSB. The Cm curve shows a broad maximum at a temperature T**, which is 30% above Tf, as expected for conventional SG systems. Our results show that even in the semiclassical regime (weak Ht), the presence of RF changes this scenario completely. Cm still shows the broad maximum at T** that is weakly dependent on &Delta. However, the Tf decreases as &Delta increases, enhancing the ration T**/Tf. In addition, the divergence in &chi3 is replaced by a rounded maximum at a temperature T* which becomes increasingly higher than Tf as &Delta enhances. As a consequence, the paramagnetic PM 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 (Tf<T<T*).

The unfolding of the PM phase described above enforce the proposal of an intermediate Griffiths phase between the PM and SG [A. Biltmo and P. Henelius, Nat. Comm. 3, 1857 (2012)]. In the quantum regime (strong Ht) , Tf decreases by increasing Ht towards a quantum critical point. In particular, we use a relationship between Ht and &Delta (&Delta=&Delta(Ht)) which reproduce qualitatively some findings of LiHoxY1-xF4 as the &chi3 behavior and the deviation of the conventional relationship between the Tf, T* and T**.