Material that changes solubility when electric potential is applied
This is a tentative that perhaps does not fit with restrictions posed by the setup of the question, but does offer an example of a class of materials whose solubility (integrity, to be more precise) is easily altered by an electrochemical process.

A number of conjugated polymers can be deposited atop an electrode by
A: electrochemical polymerisation ( this resulting in a film of the polymer in a more or less conductive state ) or

B: as pristine material if available and soluble in some solvent ( spin coating, dipping, dr blading. ..).

The state of the polymer can be electrochemically controlled by the applied potential. ( however the complete neutralisation to its insulating/semiconducting state is generally difficult ).

Overoxidation of these films results in chemical damage associated with removal of the material, directly or by subsequent washing.

3-methylthiophene and poly(3-hexylthiophene) are well known examples for an electrochemical depositable polymer and for a polymer which can be directly cast from its solutions. 

In both cases the poly(3-alkylthiophene) (P3AT) film is insoluble in polar solvents and electrolyte solutions. Applying an electrochemical potentially higher than, say, 2 V, overoxidize it damaging the skeleton to such an extent that the material fall out the electrode and / or can be washed out.

Specifically to the question: only case B is viable. 

Looking at the figure in the Q, bottom, the orange part will be consisting of neutral P3AT.

Resolution will be not too precise as overoxidation will take place at the edge of the electrode, too.