I am not familiar with work permit related sponsorship. If Canada does have some sort of dependent VISA for persons obtained work permit, your mom will provide CIC an explanation letter to explain the mistake she made on her visitor visa application, plus any backup documents to back up that in fact that was a mistake, such as you said that your mom's passport claimed she was in fact married, before she submitted her application, and such. The good thing is eventually your mom claimed correctly on her work permit application, that too can be a back up document.documents to prove their relation or just the regular requirement for sponsoring?
Anyway, good luck.