Does the current of an electrolytic cell increase if the surface area of the Cathode is increased
Just remember two major rules in electrolytic electrochemistry:


If you fix voltage, current cannot be controlled. This is called constant potential electrolysis
If you fix the current, voltage cannot be controlled. This is called constant current electrolysis


Now your electrolysis cell has resistance:

(i) It will depend on electrolytes

(ii) Surface area of the electrodes

(iii) Distance between the electrodes

So let us know your results and findings. Plot a graph systematically by choosing condition 1 or 2.