...we have this from the Persuasion FAQs:

IS MRS. CLAY WIDOWED OR DIVORCED?

Jane Austen does not say, though it is presumed that Mrs. Clay is a widow as she would be beyond the pale as a divorced woman. Moreover, her ex-husband would have had custody of the children.

Also, if Mrs. Clay's husband had abandoned her, she would certainly not be able to remarry, and I believe that even if he had, there was some social stigma on the abandoned *wife*, who would not feel comfortable socializing too publicly.