The correct option is D so that
The subordinate clause ("our children will have better lives") depends on the main clause ("We work hard") for its full meaning. In the given statement, the subordinate clause is the purpose or reason for which something is done "working hard". In order to provide a better life for our children, we have to work hard. The underlined word 'since' refers to a cause, and not the purpose. 'Purpose' and 'cause' can be confused for each other, though there's a difference between these two. 'Purpose' refers to the reason for which something is done, created or for which something exists, whereas 'cause' is the justification or explanation behind an action or event. As 'since' is a cause, and the sentence is purpose-related, hence it needs to be replaced and option A is incorrect. 'So that' is a conjunction used for purpose-related sentences, like in the given one, it refers to the purpose behind working hard. Hence it's appropriate for the sentence and option C is correct. 'Because' is a conjunction for cause-related sentences, as the given sentence does not relate to cause, but purpose, option B is incorrect. 'For' is an Illative conjunction, used when one clause is inferred from or concluded by the other in a sentence. It's a type of coordinating conjunction used when the clauses in a sentence are independent on their own, whereas in the given sentence, one clause is dependent, hence 'for' is not the appropriate option, D is incorrect.