In American English, “work something out” and “work through something” have overlapping fields of meaning.
“Work something out” frequently means to resolve something.
I am serious about working this dispute out without lawyers and without jeopardizing our relationship
indicates a desire to end a dispute amicably.
“Work through something” frequently means to analyze something carefully with a view to understanding.
I am trying to work through Ricardo’s theory of value, but am finding it quite obscure
indicates a desire to fully understand that theory.
However, we frequently want to understand something in order to resolve it. In such situations, either locution may spring to mind.
I am trying to work through this notice of a tax deficiency
with the unstated purpose of resolving it
Yes, I am studying the notice of tax deficiency. I want to work this out promptly
Either phrase may be appropriate in such a situation. They have different primary meanings, but those meanings are often closely related.