This is common, as the MLS auto populates, size or square footage
It is up to the Buyer to verify if the Addition was Completed and Finalized.
Otherwise the Buyer has likely Paid more.
In terms of the getting the additional space legalized, I would go to the city and ask them. In my experience, you may have to submit a sketch drawing for the addition, pay a fee and pass an inspection to get it done.
But, if you are uncomfortable with the the addition and the extra work needed to get it approved by the city, walk away. There will be other deals that will work better for you.
The appraiser will note that the property has additional space and no permits.
If you are obtaining financing, this issue will need to be resolved prior to your closing.
The underwriter, upon review of the appraisal, will let you know what needs to be accomplished for this.