I hope this might help:
1. Noun clause reduced by using only gerund or infinitive?
While he admitted that he had received the stolen jewelry, he denied taking part in the robbery.
While he admitted having received the stolen jewelry, he denied taking part in the robbery.
Gerund (or present participle construction) "having received the stolen jewelry" in place of a nominal that-clause "that he had received the stolen jewelry"
2. Relative clause reduced by using only participle?
My sister is coming on the train which is arriving at Platform 2.
My sister is coming on the train arriving at Platform 2.
Relative clauses function as adjectives of nouns, postmodifying train, in this case. Defining relative clause "which is arriving at Platform 2" is reduced to present participle construction (or gerund) "arriving at Platform 2"
3. Adverb clause reduced by using gerund, past participle and infinitive?
With the same example as in 1:
While he admitted having received the stolen jewelry, he denied taking part in the robbery.
Admitting having received the stolen jewelry, he denied taking part in the robbery.
"Admitting having received the stolen jewelry" is a gerund construction functioning as a clause of consession.
I'm not a teacher (yet). I took some examples from the book Longman English Grammar by L. G. Alexander.