The phrase "unprosperous marriage" does not suggest widowhood to me--unless, perhaps, he died and left her nothing. This had not previously occurred to me. The further comment "with the additional burthen of two children" would then make sense (aside from being merely humorous).

I had always assumed that Mrs. Clay was intentionally presented thus, as perhaps a wee bit beyond the pale, due to the stigma attached at that time to an unsuccessful marriage.

JA is certainly ambiguous enough about Mrs. Clay's past to lead us to believe she is not quite the sort we would want for a neighbor, yet without any proof of wrongdoing. Of such things are prejudices born, but on the other hand, sometimes you really have to go with your gut instincts.