Tom lost his keys (while) walking through the park. (Tom lost his keys while he was walking through the park.)
She left the room singing happily. (She left the room as she was singing happily.)
When I search "reduced adverbial clauses", Sentences that start with a reduced adverbial clause are usually what I find in most of the English learning articles.(E.g., "After doing something, ..." , "Being late, ...")
The two sentences quoted above do not start with a reduced adverbial clause and they do not have a comma in each of them, and the article above even put a complete adverbial form of each sentence behind them. I wonder if their present participle parts are in fact a type of reduced adverbial clauses.