Suggested answer:
(1) Cure is found only in combination with for:
...she had a cure for it...
Damage combines only with to:
There might have been some damage to the rear bumper...
Invitation is only found with to, although on one occasion it is followed by on, albeit not with a postmodifying function.
...tearing up an invitation to the Geelong Grammar School dance...
(2) Opinion about, opinion of and opinion on are all examples of noun + preposition combinations where the PP has a modifying function:
'What's your opinion about these forgeries?'
I don't think Bobby had a very high opinion of me.
...voicing an intolerant opinion on the Jesuits.
(3) An example of opinion followed by a preposition where the PP does not have a modifying function in the NP:
...she was sensitive to public opinion in the village...
(4) The PP following opinion in the example under (3) functions as an Adverbial of place. The PPs under (2) have a postmodifying function in the NPs headed by opinion.
Read about noun + preposition combinations here
Terminology: preposition, prepositional phrase (PP), adverbial, postmodifier, Noun Phrase (NP)