There are four distinct constructions which employ HAVE with a past participle and an object. One is the perfect construction, which I assume you are familiar with; the others I'll call the causative, the experiential and the attained-state constructions.
Below I use the conventional form VERBen for the past participle.
THE PERFECT CONSTRUCTION
I have explained it to you. –I performed the explanation.
Demonstrators have thrown eggs at the minister. –The demonstrators performed the throwing.
In this construction, SUBJECT is the AGENT of VERB, and OBJECT typically lies after the past participle.
SUBJECT HAVE VERBen OBJECT
In the other three constructions the OBJECT of VERB lies between HAVE and VERBen, and SUBJECT is the AGENT only of HAVE, not of VERB.
SUBJECT HAVE OBJECT VERBen
THE CAUSATIVE CONSTRUCTION
I will have it explained to you (by my secretary).
The opposition had eggs thrown at the minister (by paid demonstrators).
- In this construction, SUBJECT is the AGENT of HAVE in the sense "induce, cause", but the action of VERBen is carried out by some other party: SUBJECT causes that other party to perform the action. VERBen is in effect a passive participle; if the AGENT of VERB is to be named that must be done with a by phrase, exactly as in the passive construction.
THE EXPERIENTIAL CONSTRUCTION
1a. You've had it explained often enough.
- In this construction, SUBJECT undergoes the experience of OBJECT being VERBen; again, VERBen is a passive participle and its action is carried out by some other party. Distinguish this from the perfect (it was not you who explained) and from the causative (you didn't cause somebody else to explain it) —You have undergone the experience of its being explained.
1b. The minister had eggs thrown at him.
- Distinguish this from the perfect (it was not the minister who threw the eggs) and from the causative (the minister didn't incite somebody else to throw eggs at him)—The minister underwent the experience of eggs being thrown at him.
THE ATTAINED-STATE CONSTRUCTION
We now have the problem solved.
Longman's description is I think misleading. In this construction HAVE retains its core lexical sense of "hold, possess, receive", and the past participle is basically an adjectival object complement: We have the problem [and it has been] solved or We have the problem [and it is] in a solved state. VERBen is emphatically 'perfective', it names a state which OBJECT has attained—which is probably the reason for Longman's description.
This construction in fact appears to be the oldest of the four constructions, present in both Germanic and Romance languages—even Latin—and it is widely conjectured to be the origin of the perfect construction. Today the attained-state construction is used mostly to introduce the fact of OBJECT's VERBen state as a launching point for describing what can be done next: "We have the problem solved, so we can move ahead to ...".