The paper investigates the quasi steady state flow of viscoelastic mucus and air in a circular two-layered mathematical model under a time-dependent pressure gradient. The model resembles the trachea in the human respiratory system in the context of bronchial constriction during a slight cough. Within this paradigm, air is viewed as an incompressible Newtonian fluid, whereas mucus is thought of as a viscoelastic Voigt element, with both viscous and elastic properties. The conclusions of the study show that the transportation of mucus is dramatically decreased for a given pressure drop as the mucus’s elastic modulus increases, demonstrating how higher elasticity might impede mucus flow. Furthermore, the study shows that when mucus viscosity rises, mucus transport decreases even more, especially in narrowed or restricted airways. This effect is amplified in different pressure drops. It is interesting to note that, in spite of these modifications in mucus dynamics, air travel stays constant in all cases, emphasizing the unique and diverse effects of bronchial constriction on mucus versus air movement in the respiratory system.