will the autoionization of water reduce due to common ion effect?
Adding acid to the water will not stop the auto-ionization of the water, so: the auto-ionization will continue to give equal concentrations of hydroxide and hydronium. But the addition of acid to the water increases the concentration of the hydronium ions in the solution, pushing water balance with its ions to the left to resist the increase in the concentration of hydronium and reach a new equilibrium. The concentrations of hydronium and hydroxide produced by auto-ionization of water are reduced, where the concentrations of $\ce{H_3  O^+}$ or $\ce{OH^-}$ from water ionization is less than$\pu{ 10^{-7}}$. 

$$\ce{{[H_3O^+]}_\text{(after acid  addition)}=\ce{[H_3O^+]}_\text{(produced by acid dissociation)} +\ce{[OH^-]}_\text{(produced by water auto-ionization )}}$$