A challenging part of the question of how elements heavier than iron are created in extreme, astrophysical environments is the creation of p-isotopes. The lack of needed nuclear data presents an obstacle in nailing down the precise site and astrophysical conditions for the production of these isotopes. The p-isotope $^{92}$Mo represents one of the most severe cases of underproduction. The main destruction mechanism of this isotope in the standard description of the p-process is through the $^{92}$Mo($\gamma$, p)$^{91}$Nb reaction. Measurements on the nuclear level density and $\gamma$ strength function of $^{92}$Mo have been carried out at the Oslo Cyclotron Laboratory. TALYS cross section and reaction rate calculations using the experimental results as input are presented, providing constraints on the $^{91}$Nb(p, $\gamma$)$^{92}$Mo (and consequently the inverse) reaction rate. Further, the reaction rates extracted in this work were used in network calculations for the scenario of a p-process taking place in a type II supernova explosion as the shock front passes through the O-Ne layer of a 25 solar mass star. We conclude that there is no salvation in the nuclear input alone in the $^{92}$Mo underproduction problem, strengthening previous conclusions pointing towards more exotic astrophysical scenarios.