  We have studied in detail the $M$-$p$ balanced spin glass model, especially
the case $p=4$. These types of model have relevance to structural glasses. The
models possess two kinds of broken replica states; those with one-step replica
symmetry breaking (1RSB) and those with full replica symmetry breaking (FRSB).
To determine which arises requires studying the Landau expansion to quintic
order. There are 9 quintic order coefficients, and 5 quartic order
coefficients, whose values we determine for this model. We show that it is only
for $2 \leq M < 2.4714 \cdots$ that the transition at mean-field level is to a
state with FRSB, while for larger $M$ values there is either a continuous
transition to a state with 1RSB (when $ M \leq 3$) or a discontinuous
transition for $M > 3$. The Gardner transition from a 1RSB state at low
temperatures to a state with FRSB also requires the Landau expansion to be
taken to quintic order. Our result for the form of FRSB in the Gardner phase is
similar to that found when $2 \leq M < 2.4714\cdots$, but differs from that
given in the early paper of Gross et al. [Phys. Rev. Lett. 55, 304 (1985)].
Finally we discuss the effects of fluctuations on our mean-field solutions
using the scheme of H\"{o}ller and Read [Phys. Rev. E 101, 042114 (2020)}] and
argue that such fluctuations will remove both the continuous 1RSB transition
and discontinuous 1RSB transitions when $8 >d \geq 6$ leaving just the FRSB
continuous transition. We suggest values for $M$ and $p$ which might be used in
simulations to confirm whether fluctuation corrections do indeed remove the
1RSB transitions.
