I know newspapers use short syntax to get attention of the readers but can we use the bold form is it grammaticaly correct?
And in the 4th example normally does it need 'about' after 'backs off' and an s for Obama's birth claim ?
1.Hillary Clinton to return to campaign trail Thursday
2.Barack Obama, Bill Clinton To Fill In For Hillary Clinton
3.Hillary Clinton to Address Millennial Voters During Campaign Event in Philadelphia
4.Trump finally backs off Obama birth claim
Newspaper headlines have their own unique style. While it is acceptable within this context, it would be horribly wrong to use it elsewhere, such as in a conversation.
Wikipedia describes what you are seeing in the first three headlines:
The future is expressed as "to" followed by a verb, e.g. "Governor to sign bill".
As for the last headline, you're seeing something ubiquitous. Nouns can become adjectives readily, as "Obama" has:
In fact, the alternative you suggest is not used; the occurrences are negligible when compared with the original phrasing.
According to grammarbook.com:
Beware of false possessives, which often occur with nouns ending in s. Don't add apostrophes to noun-derived adjectives ending in s. Close analysis is the best guide.
Incorrect: We enjoyed the New Orleans' cuisine.
In the preceding sentence, the word the makes no sense unless New Orleans is being used as an adjective to describe cuisine. In English, nouns frequently become adjectives. Adjectives rarely if ever take apostrophes.
Incorrect: I like that Beatles' song.
Correct: I like that Beatles song.Again, Beatles is an adjective, modifying song.
Incorrect: He's a United States' citizen.
Correct: He's a United States citizen.
Newspaper headlines typically use an abbreviated grammar for their headlines if space is tight.
Since use of space is pretty strictly controlled for typography reasons, editors have to get the meanings of the stories into a specific physical width (for example, the text is going to be x inches tall by 5 inches wide to fit into the layout of that page). This is why some words are dropped, and some words are kept.
