The police arrested him for speeding.

"The police" = S
"arrested" = V
"him" = O
"(F)or speeding" is a reason- and an adverbial clause?

In my book, a clause contains a verb or verbs. "(S)peeding " is a gerund, thus is a noun.
How is it?

Question #2:

1. "(F)or speeding" is a reason clause and an adverbial clause?
2. "(F)or speeding" is a reason- and an adverbial clause?

2. = 1. ?

Do you agree with my hyphen to follow "reason"?
For speeding is a prepositional phrase functioning as an adverbial.

No hyphen.
Thanks MM.
Why still I read clauses has a verb or more, when they more often than not do not?
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Please give me an example of a clause without a verb, IK.
That's interesting, IK. The verb is ellipted:

hence [came/we see/etc] the capital importance of arabesques, of geometrical and botanical decorative motifs.

In parsing, it remains a clause with an understood verb, I think, though on paper we are left with nothing but an adverb and a noun phrase. Still, clauses without overt verbs are rare.
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Because there are so many different ways to analyze such structures, I am always hesitant to answer such questions. One never knows whether the student is studying from a book which takes a "traditional" approach to grammar and its terminology or from a more "transformational grammar" approach. I have come to find the transformational approach the more insightful approach over the years, so I find it difficult not to see phrases and clauses mostly from that point of view.

That caveat announced, I see for speeding as a (transformed) adverbial clause. It is a transformation of the underlying for he was speeding. The transformation raises a verb to a noun. (Not much of a transform: was speeding > speeding.) The subject of the underlying clause (he) and the preceding object (him) in the main clause are coreferential, so the subject of the subordinate clause can be dropped. The result in the surface structure is the prepositional phrase for speeding.

It turns out that, from the point of view of transformational grammar, the so-called small clauses are all verbless.

We painted the house white. (the house white is called a small clause in this non-traditional analysis.)