You seem to be confusing dashes -- which are used to separate text for empahsis, to change the flow, or to add additional information as I am here -- with hyphens. (I can't seem to make a dash in here so I am using two hyphens together.
In your sentence, These... Tabby is an independent clause, and "often... situation is aslo an independent clause. To separate them, you cannot use a comma. You need either a period/full stop, a semi-colon, a conjunction, or... a dash!
Use use of the hyphen is to join two (or more) words that work together to modify an noun.
So, your final product:
These were rough years for Kind and Tabby -- often, their child Naomi would get ear infections, which did not help their [no comma] already-poor financial situation.
Student or Learner