The relevance of this research lies in the need to develop scientifically based methods for calculating and designing a transmission shaft with a hardened coating of increased strength and service life of a core drilling pump drive that can allow for a redistributing of resistance forces along the contact surfaces of the gear. This relevance is confirmed by the need to improve domestic methods for designing drive shafts of increased reliability which can ensure the development of frozen soils during deposits exploration. The purpose of the research is to increase the energy efficiency and service life of the high-loaded drive gear teeth of core drilling pump transmission shafts by justifying the critical loads and stresses in hardened gear coatings acting under intense wear of the contact surface with a broken contact symmetry. The criteria for the effective wear area with an uneven contact cross-section at the maximum bending moments of the transmission shaft of the drilling pump were justified and presented in the work. Additionally, the process of interaction of the transmission shaft gear teeth with the eccentric shaft gear at uneven axial torques was investigated. The effective power (Ng) of the gearing of the drive transmission shaft gear and the eccentric shaft gear, which characterizes the energy consumption of the drill bit depth stroke, was justified. This work also proposes a method of substantiating the technological and power parameters of the transmission shaft by using Legendre polynomials. A nomographic chart was developed for the determination of the dependence of the contact stress base cycles on the change in the load distribution factor and the contact spot deviation factor from the design axis λ.