In this paper, the proposed dark energy, Tsallis holographic dark energy (THDE), infrared cut-off with the Hubble horizon has been investigated in the Bianchi-I (axially symmetric) anisotropic model with a hybrid expansion law. It has been observed that the THDE is in tune with the accelerating Universe with equation of state (EoS) parameter ($\omega_{T}$ < $−1/3$) in the $k$-essence region. We have used the statefinder diagnostic in our model. In addition, we try to accommodate the perspective of dark energy by the avenue of reconstructing the evolution of scalar field potential. We have considered the $k$-essence for the analysis of this reconstruction, showing the accelerated expansion at present.