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Zinc tungstate is a perspective scintillation material for the cryogenic scintillating bolometers. High energy resolution, low background level and discrimination between different types of particles makes it suitable for the registration of the rare processes such as double beta decay, neutrinoless double beta decay and weakly interacting massive particles. In this work the energy transfer processes to the emission centers is studied in ZnWO4 and ZnWO4:Mo single crystals in the temperature range 10-300 K. Nominally undoped ZnWO4 was grown by the low temperature gradient Czochralski technique. ZnWO4:Mo single crystal was grown by the conventional Czochralski method. The temperature dependence of the energy transfer to the intrinsic emission centers in ZnWO4 is connected with the modification of the Onsager sphere radius, the numerical simulation describing the process is presented. The competitive role of the radiative relaxation channel related to the attendant impurity of molybdenum is studied in ZnWO4:Mo. It is shown that below 60 K the energy transfer of the interband excitations to the MoO6 emission centers is terminated due to the localization of the holes at WO6 complexes.