We analyze 9 years of PASS 8 $textit{Fermi}$-LAT data in the 60$-$500 MeV range and determine flux upper limits (UL) for 17 gamma-ray dark pulsars as a probe of axions produced by nucleon-nucleon Bremsstrahlung in the pulsar core. Using a previously published axion decay gamma-ray photon flux model for pulsars which relies on a high core temperature of 20 MeV, we improve the determination of the UL axion mass ($m_a$), at 95 percent confidence level, to 9.6 $times$ 10$^{-3}$ eV, which is a factor of 8 improvement on previous results. We show that the axion emissivity (energy loss rate per volume) at realistic lower pulsar core temperatures of 4 MeV or less is reduced to such an extent that axion emissivity and the gamma-ray signal becomes negligible. We consider an alternative emission model based on energy loss rate per mass to allow $m_a$ to be constrained with $Fermi$-LAT observations. This model yields a plausible UL $m_a$ of 10$^{-6}$ eV for pulsar core temperature $<$ 0.1 MeV but knowledge of the extent of axion to photon conversion in the pulsar $B$ field would be required to make a precise UL axion mass determination. The peak of axion flux is likely to produce gamma-rays in the $leq$ 1 MeV energy range and so future observations with medium energy gamma-ray missions, such as AMEGO and e-ASTROGAM, will be vital to further constrain UL $m_a$.
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