Legal Question in Landlord & Tenant Law in California

Without knowing what is illegal about the construction I can't answer your question. If a living unit is habitable, you can't get out of paying rent or moving out before your rental term is up just because it was built without permits or in violation of a zoning or use permit restriction, unless the city or county required the landlord to remove you from the unit because they found out about the code, zoning or use permit violations. There are literally thousands of "in-law" units that are in technical violation of building permit, building code, zoning our use regulations, but are perfectly habitable and safe. Most communities turn a blind eye to the violations because they increase the stock of affordable housing, as long as the units are habitable and safe.

Also, regardless of the legalities, if a unit is not legally habitable, and the landlord offers a habitable unit in exchange, you certainly have absolutely nothing to sue about at that point.