Graphene/hBN heterostructures are promising active materials for devices in the THz domain, such as emitters and photodetectors based on interband transitions. Their performance requires long carrier lifetimes. However, carrier recombination processes in graphene possess sub-picosecond characteristic times for large non-equilibrium carrier densities at high energy. An additional channel has been recently demonstrated in graphene/hBN heterostructures by emission of hBN hyperbolic phonon polaritons (HPhP) with picosecond decay time. Here, we report on carrier lifetimes in graphene/hBN Zener-Klein transistors of ~30 ps for photoexcited carriers at low density and energy, using mid-infrared photoconductivity measurements. We further demonstrate the switching of carrier lifetime from ~30 ps (attributed to interband Auger) down to a few picoseconds upon ignition of HPhP relaxation at finite bias and/or with infrared excitation power. Our study opens interesting perspectives to exploit graphene/hBN heterostructures for THz lasing and highly sensitive THz photodetection as well as for phonon polariton optics.