I'm not aware of an appeal process for sponsor side. This would probably be a situation to discuss with your lawyer since they gave you the poor advice to file when the sponsor is on welfare
OR better yet, ditch the lawyer, send the money to your wife to help her bridge the gap and earn an income... There are plenty of govt funded programs that she can take advantage of that are NOT considered assistance, such as child care subsidy and federal and provincial child tax credits... She might need help while they kick in, but life is manageable and can be geared to income if your income is low...
Good luck