I am not sure if "specify" is the right word here, so please edit it if there is a more suitable expression.
So I know that noun phrases can be "specified" by dependent clauses, like this,
The cat that was lying there woke up.
In which that was lying there is a dependent clause that "specifies" cat. But can you do the same thing with the verb phrase/verb?
Yes, of course.
The cat that was lying there woke up with a satisfied yawn.
Updated: I am adding to this based on some comments.
In my revised version of the example sentence, the verb is woke. It is modified by with a satisfied yawn which acts adverbially—it describes how the cat woke up. In the same fashion, that was lying there acts adjectivally to modify the noun cat.
There is a slight difference between the two however. That was lying there is a relative clause, whereas with a satisfied yawn is not.
I can think of no way of modifying the example sentence that produces an adverbial relative clause after woke up, but I can think of an alternative sentence:
I didn't know who would be there.
Here, the verb is know and who would be there is both a relative and adverbial clause that modifies it.
