Element doping can break the crystal symmetry and realize the topological phase transition in quantum materials, which enables the precise modulation of energy band structure and microscopic dynamical interaction. Herein, we have studied the ultrafast photocarrier dynamics in Zn-doped 3D topological Dirac semimetal ${\mathrm{Cd}}_3{\mathrm{As}}_2$ utilizing time-resolved optical pump-terahertz probe spectroscopy. Comparing to the pristine ${\mathrm{Cd}}_3{\mathrm{As}}_2$, we found that the relaxation time of the lightly doped alloy is slightly shorter, while that of the heavily doped alloy exhibits a significant prolongation. Pump-fluence- and temperature-dependent transient terahertz spectroscopy indicated that in pristine and lightly doped samples within nontrivial semimetal phase, the photocarrier dynamics are dominated by the cooling of Dirac fermions. In heavily doped alloy, however, the observed longer relaxation process can be attributed to interband electron-hole recombination, which is a result of doping-induced transition into a trivial semiconductor phase. Our investigation highlights that Zn-doping is an effective and flexible scheme for engineering the electronic structure and transient carrier relaxation dynamics in ${\mathrm{Cd}}_3{\mathrm{As}}_2$, and offers a control knob for functional switching between diverse optoelectronic devices within the realm of practical applications.