The correct answer is: E. No error
The sentence is grammatically correct. The word “that” is used to introduce a subordinate clause, and the word “whether” is used to introduce a question. In this case, the subordinate clause is “whether he would be permitted to leave early.” The question is whether the subordinate clause is necessary. It is not necessary, because the main clause already expresses a question: “he asked his Superiors.” Therefore, the word “that” can be omitted, and the sentence can be written as:
On resuming his duty, he asked his Superiors if he would be permitted to leave early.
However, the original sentence is also grammatically correct.