  The Peccei-Quinn (PQ) symmetry does not need to be exact, and even a tiny
source of PQ breaking not aligned with the QCD anomaly might have significant
phenomenological implications. In this study, we examine the effects of a
general class of PQ-breaking operators on the axion cosmological production via
misalignment, focussing on both temperature-dependent and independent
PQ-breaking potentials. In particular, we show that a variant of the trapped
misalignment mechanism can delay the onset of axion oscillation, leading to an
axion dark matter window with $m_a \gg 10^{-5}$ eV. This scenario is testable
through various experimental approaches, including standard axion haloscopes
and helioscopes, as well as searches for electric dipole moments and
axion-mediated forces.
