  The PH-Pfaffian state having 1/2 central charge is consistent with the
thermal Hall conductance measurement of 5/2 fractional quantum Hall system, but
lacks support from the existing numerical results. In this paper we propose a
new state described by a wavefunction obtained by multiplying a Hafnian factor
to a PH-Pfaffian wavefunction. We call this new state the Hafnian PH-Pfaffian
state. In spherical geometry, the Hafnian PH-Pfaffian state has the same
magnetic flux number as the Pfaffian state, allowing a direct numerical
comparison between the two states. Results of exact diagonalization of finite
systems in the second Landau level show that the overlap of the exact ground
state with the Hafnian PH-Pfaffian state exceeds that with the Pfaffian state
when the short range component of the Coulomb interaction increases to a
certain level, lending a numerical support to the Hafnian PH-Pfaffian state. We
further show the Hafnian PH-Pfaffian state is mathematically identical to the
newly proposed compressed PH-Pfaffian state [arxiv: 2001.01915 (2020)] formed
by "compressing" the PH-Pfaffian state with two flux quanta removed to create
two abelian Laughlin type quasiparticles of the maximum avoidance from one
another. As the result we argue that the Hafnian PH-Pfaffian state has the same
central charge as the PH-Pfaffian state, and is therefore consistent with the
thermal Hall conductance measurement. Finally we present numerical results on
two new wavefunctions formed by increasing the relative angular momentum by two
for each of the paired composite fermions of the PH-Pfaffian state.
