The correct option is
C If he were to continue to be sick tomorrow, he will take the day off.
Option A:
"Despite" is a preposition which is used to show something which happened although something else might have prevented it. For example: "Despite my hard work, I failed in the examination". As it is not a conjunction, this option is incorrect. Option B: "So that" is a subordinating conjunction which is used to express cause or reason. It is used to connect two sentences when one is the cause of the other. The sentence, "He will take the day off so that he can continue to be sick tomorrow" means: Taking the day off will enable him to be sick tomorrow. Such a sentence does not convey any meaning. A more appropriate sentence would be: "He will take the day off so that he can recover quickly from his illness". Hence, option A is incorrect.
Option C: "Although" is a subordinating conjunction of concession which connects two clauses which contrast each other. As the two sentences do not contrast each other, this is not the most appropriate conjunction to be used. Hence, option C is incorrect.
Option D: "If" is a subordinating conjunction which expresses a condition. It precedes a clause/event which must happen first in order for the other event to occur. The clause, "he will take the day off" is conditional upon the clause, "He were to continue to be sick tomorrow". Hence, "If" is the most appropriate conjunction to be used and this option is correct.