I think it is appropriate to take a step back and consider what grammarians understand predicates to be. I have asked a number of established syntacticians directly how they use the term predicate. The responses I have received are quite varied.
In order to understand what a “secondary” predicate is, though, I think one should first know what a “primary” predicate is. Consider the question with respect to the following sentence:
(1) Frank has smiled at Susan.
If you ask a grammarian directly what the main predicate (i.e. the primary predicate) in this sentence is, you are likely to receive one of three responses:
Response 1
The predicate is everything but the subject, hence it is has smiled at Susan.
Response 2
The predicate corresponds to the main verb, but in an abstract sense, hence the predicate in this case is SMILE, or perhaps SMILE AT.
Response 3
The predicate is the main verb and any auxiliaries that are present, thus in this case, the predicate is has smiled, or perhaps has smiled at.
Response 1 is that of traditional school grammar and one advocated by many formal semanticists. Response 2 is more likely to be produced by theoretical syntacticians. Response 3 is also encountered on occasion and is associated with the grammar tradition of the German language. I personally favor the latter.
Consider the issue with respect to a sentence involving copular be:
(2) Susan is a supporter of free speech.
Those who advocate Response 1 view is a supporter of free speech as the main predicate; those who advocate Response 2 view SUPPORTER OF as the main predicate; and those who advocate Response 3 view is a supporter of as the main predicate.
Now to understand what a “secondary predicate” is, one can say that, well, it is any predicate in the given sentence that is not the main predicate. But of course this approach to secondary predicates is reliant on which of the three responses concerning the main predicate one prefers.
To come to some of the specifics in the question, I think many grammarians understand the bolded expressions in the next sentences to be secondary predicates:
(3) Susan cooks naked.
(4) Frank scrubbed the sink clean.
(5) You consider them lazy.
The primary predicate in each of these sentences is in italics (assuming Response 3 above), and the bolded adjectives are the secondary predicates. These secondary predicates serve to assign the indicated property to a preceding argument of the main predicate. Thus, it is Susan who cooks naked, it is the sink that is clean, and it is them who are lazy. This is just one way to understand what a secondary predicate is, though. On this understanding, a secondary predicate is always such that it predicates a property of an argument of the main predicate, which is the primary predicate.
Let’s consider another type of example:
(6) Susan thinks Fred talks too much.
In this case, the primary predicate is thinks (again, assuming Response 3 above), which means that talks is secondary in a sense. But now I think many grammarians might disagree. They might claim that since talk is the main predicate inside the embedded clause, it too should be viewed as a primary predicate, but simply as an embedded one.
Concerning quickly in the sentence The building was built quickly, it is an adjunct. If one chooses to view adjuncts as secondary predications in general, then any adjunct is or contains a secondary predicate. This understanding of predicates widens significantly the scope of how one views predicate-argument structures. While a strong case for doing this can be made, it is not without its difficulties.
Concerning the term overt predicate head, my guess is that it refers to the following cases:
(7a) We consider that good.
(7b) We consider that to be good.
The overt predicate head of the secondary predicate to be good in (7b) is to be. This predicate head is omitted or elided or otherwise understood as absent from (7a).
Concerning so-called “small clauses”, much rides on the syntactic analysis. Consider the next sentence in this regard:
(8) We expect you to help.
The string you to help is viewed as a constituent small clause by many in the syntax world. Yet others would assume a flat analysis, one in which you and to help are sister constituents that do not form a greater constituent together. In any case, the secondary predicate in such cases is to help; it counts as a secondary predicate in a clear way since it clearly predicates the property of helping over the preceding you.
There are other points in the question that are difficult to address due to the vagueness of what is meant. Concrete examples illustrating what the question is would allow for commentary and clarification.
