The preposition "through" often couples with other prepositions, like "into," "across," "under," and others to convey more nuanced spatial, temporal, or contextual ideas.
In a sentence, you typically use "through" to describe a process or action, and "under" to specify the conditions or circumstances under which the action takes place.
The phrase "through with" is generally considered to be neutral in tone. It is widely accepted in both spoken and written English across various contexts.