No sooner he had returned home then his mother felt happy.

The correct answer is D. No improvement.

The sentence "No sooner he had returned home then his mother felt happy" is grammatically correct and uses the correct idiom. The idiom "no sooner...than" means "as soon as." In this case, the sentence is saying that as soon as the man returned home, his mother felt happy.

The other options are all incorrect. Option A, "had he returned home when," is not idiomatic. The correct idiom is "no sooner...than." Option B, "he had returned home than," is the same as the original sentence, but it is not necessary to repeat the word "he." Option C, "did he return home than," is not idiomatic. The correct idiom is "no sooner...than."

Therefore, the correct answer is D. No improvement.