  We consider a higher order term in the $\delta N$ expansion for the CMB power
asymmetry generated by a superhorizon isocurvature field fluctuation. The term
can generate the asymmetry without requiring a large value of $f_{NL}$. Instead
it produces a non-zero value of $g_{NL}$. A combination of constraints lead to
an allowed region in $f_{NL}-g_{NL}$ space. To produce the asymmetry with this
term without a large value of $f_{NL}$ we find that the isocurvature field
needs to contribute less than the inflaton towards the power spectrum of the
curvature perturbation.
