Hello, Beth
Believe it or not, that type of clause actually exists! ~chuckles~
Saying that a reduced clause is non-finite, however, is redundant. It is a characteristic of reduced clauses, both adverbial and relative, to be non-finite (meaning that they don't have a conjugated or 'finite' verb).
On the other hand, not all finite clauses are reduced clauses.
Reduced adverbial clauses commonly appear at the beginning of the sentence and are separated from the rest of the sentence by a comma. These clauses are either present participial (-ing) or past participial (-ed) constructions. They are called 'reduced' clauses because they are precisely the 'reduced' version of a finite clause.
Here are a few examples of reduced adverbial clauses (in parentheses):
These are present participial clauses.
"(Walking down the street), we passed by a flower shop."
"(Our teacher being absent), we left the classroom."
They can be expanded into finite clauses:
"(While/When/As we were walking down the street), we passed by a flower shop."
"(Since/Because/As our teacher was absent), we left the classroom."
These are past participial clauses.
"(Appalled by the sad news), the woman started to cry."
"(The bill having been paid), I buttoned up my coat to leave."
The corresponding finite clauses could be:
"(Since/Because/As she was appalled by the sad news), the woman started to cry."
"(After/As soon as the bill had been paid), I buttoned up my coat to leave."
Students as well as native speakers of English tend to make mistakes in this type of clause when they produce sentences in which the clause appears as what is called a 'misrelated participle'.
Here's an example I love:
"Walking up the stairs, my pen fell off my pocket."
According to that sentence, the pen, not I, was walking up the stairs. In order for this type of clause to be used correctly, IF the clause itself doesn't have an explicit subject, it will be understood that the missing subject will be the same as that of the sentence (as in the two first examples).
I hope it helps.
Miriam