This is an instance of HAVE employed in a resultative sense—it’s the sense at OALD 22:
22 (used with a past participle) have something done to suffer the effects of what somebody else does to you
She had her bag stolen.
The use is actually broader than OALD suggests; it as also employed to express positive results:
Now that we have that settled we can move on to the next item.
This use goes all the way back to Old English, and was in fact the ancestor of the Perfect constructions which emerged in Middle English. In Old English the underlying syntax at that time was HAVE + [Direct Object] + [PaPpl], with the PaPpl modifying the Direct Object. (We know this because nouns and adjectives had case endings in OE, and in this construction the PaPpl 'agrees' with the Direct Object). In Modern English, however, it is probably best understood as a reduced subordinate clause:
He had [ [that] a curse [was] put on him. ]
I advance this parsing because in ModE there is also a use with a bare infinitive instead of a past participle; this is employed when the verb in question may be used both transitively and intransitively and the participle would be ambiguous:
She had her house burn down. ... which may be similarly parsed as She had [that] her house burned down.
Using active burn instead of passive burned here prevents this being parsed as transitive burned down used with causative HAVE, as in OALD's 23:
23 (used with a past participle) have something done to cause something to be done for you by somebody else
You've had your hair cut!
We're having our car repaired.
This use also supports use with an explicit subject, OALD's #24:
24 to tell or arrange for somebody to do something for you
• have somebody do something
He had the bouncers throw them out of the club.