A proposal of the existence of an Anomalous phase (A phase) [Das et al., Phys. Rev. Lett. 131, 056202 (2023)PRLTAO0031-900710.1103/PhysRevLett.131.056202] at the experimental range of moderate Landau-level-mixing strength has recently been made for the 5/2 state. We here report that the gapped A phase is generic to the sequence of spin-polarized fractional quantum Hall states with filling fractions ν=n/(nm-1) and ν=1-n/(nm-1), (n≥1,m≥3), that exhausts almost all the observed states and also predicts some states in the second Landau level for GaAs systems. Our proposed trial wave functions 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(π^{2}k_{B}^{2}T/3h) for all the states in these sequences.