In a cell that utilizes the reaction Zn(s)+2H+(aq)→Zn2+(aq)+H2 (g) addition of H2SO4 to cathode compartment, will
Zn(s)+2H+(aq)⇌Zn2+(aq)+H2(g)
Ecell=E∘cell−.0592log[Zn2+][H+]2
When H2SO4 is added then [H+] will increase therefore Ecell will also increases and equilibrium will shift towards right.