Disclaimer: The materials provided below are informational and should not be relied upon as legal advice.
Your description does not state why the landlord is giving you a notice to vacate. If your only agreement with the landlord was your verbal agreement to pay $600 per month, you are a month-to-month tenant. Absent any local rent control ordinances (I am not familiar with Riverside), a landlord would be able to terminate a month-to-month tenancy with a properly served 30-day notice, or a 60-day notice if you have been residing there for over 2 years. Generally, the law of fixtures provides that any work performed by a tenant becomes a part of the landlord's property, unless you had made a prior agreement with the landlord to compensate you for the work. If the property was in fact uninhabitable when you moved in, you could at that point have an uninhability defense to nonpayment of rent, but it seems like you have already brought the property to habitable conditions. Be sure to consult your own attorney to protect your legal rights.
The short answer is yes, the tenant can seek compensation if the tenant can prove the oral agreement.
However, the tenant must file a lawsuit that is separate from any unlawful detainer lawsuit. An unlawful detainer is a summary proceeding. As such, an unlawful detainer proceeding is given legal precedence over all other civil actions so that it can be quickly heard and determined. (Code of Civil Procedure §1179a.) In other words, the tenant could not and should not file a cross-complaint in the unlawful detainer lawsuit, but rather file a separate small claims or superior court civil lawsuit. A cross-complaint is not permitted in an unlawful detainer lawsuit. A defendant has no right to file a cross-complaint or counterclaim because possession is the only issue being litigated in an unlawful detainer action. (Vella v. Hudgins (1977) 20 Cal.3d 251, 255.)
Frank W. Chen has been licensed to practice law in California since 1988. The information presented here is general in nature and is not intended, nor should be construed, as legal advice for a particular case. This posting does not create any attorney-client relationship with the author. For specific advice about your particular situation, consult with your own attorney.