  A proposal of the existence of an {\em Anomalous} phase ($\mathcal{A}$ phase)
[https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.131.056202 Das et
al., Phys. Rev. Lett. 131, 056202 (2023)] at the experimental range of moderate
Landau-level-mixing strength has recently been made for $5/2$ state. We here
report that the gapped $\mathcal{A}$ phase is generic to the sequence of
spin-polarized fractional quantum Hall states with filling fractions $\nu =
n/(nm-1)$ and $\nu = 1-n/(nm-1)$, $(n \geqslant 1,\,m\geqslant 3)$, that
exhausts almost all the observed states and also predicts some states in the
second Landau level for GaAs systems. Our proposed trial wavefunctions for all
these states have remarkably high overlaps with the corresponding exact ground
states and can support non-Abelian quasiparticle excitations with charge
$e/[2(nm-1)]$. By analyzing edge modes, we predict experimentally verifiable
thermal Hall conductance $2.5(\pi^2 k_B^2T/3h)$ for all the states in these
sequences.
