The two-dimensional Ising model on a distorted kagome lattice is studied by means of exact solutions and the tensor renormalization-group (TRG) method. The zero-field phase diagrams are obtained, where three phases such as ferromagnetic, ferrimagnetic, and paramagnetic phases, along with the second-order phase transitions, have been identified. The TRG results are quite accurate and reliable in comparison to the exact solutions. In a magnetic field, the magnetization (m), susceptibility, and specific heat are studied by the TRG algorithm, where the m=1/3 plateaux are observed in the magnetization curves for some couplings. The experimental data of susceptibility for the complex Co(N3)2(bpg)$·$DMF4/3 are fitted with the TRG results, giving the couplings of the complex J=22 K and J$′$=33 K.