Grammaticality of 'that' relative clause in phrase beginning with possessive pronoun [duplicate]

Whether your examples are acceptable arguably depends on the context.

Let me concentrate on the second example:

[2] ^?They scrapped our project that had taken us so much effort.

The issue isn't really a restrictive vs nonrestrictive relative, but rather integrated vs non-integrated relative. And while integrated relatives are usually restrictive, they aren't always; this is what the examples in [18] and [19] show (see below).

(Also note that the version with the definite article is definitely acceptable: They scrapped the project that had taken us so much effort.)

While [2] is of uncertain acceptability, the following very similar example is perfectly acceptable:

[3] They scrapped our project that had generated so many losses.

Here the relative clause does not serve to restrict anything. Instead, the reason it is integrated is that it is an essential part of the reason why the project was scrapped.

In fact, I am pretty sure there could be contexts in which [2] would be acceptable. What would be required is that our taking so much effort is somehow essentially connected to why the project was scrapped.

Discussion

Consider the following passages from CGEL:

[16]  They only take in overseas students who they think have lots of money.
(The people referred to by they do not take in all overseas students, but only those from the subset they believe to have lots of money)

[17]  They only take in overseas students, who they think have lots of money.
(This time the relative clause does not pick out a subset of overseas students, but makes an assertion about overseas students in general.)

Contrasts like these provide the basis for the traditional classification of relative clauses as 'restrictive' ([16]) and 'non-restrictive' ([17]). We prefer to distinguish the two classes as integrated vs supplementary because there are many places where the contrast is not a matter of whether or not the relative clause expresses a distinguishing property.

[18]  The father who had planned my life to the point of my unsought arrival in Brighton
         took it for granted that in the last three weeks of his legal guardianship I would still act as he
         directed.

The narrator is three weeks short of eighteen and is saying that his father took it for granted that during those three weeks he would continue to do as his father directed. The relative clause here belongs to our integrated class: it cannot be omitted or spoken on a separate intonation contour and allows that as an alternant of who (albeit somewhat less favoured). Yet it does not serve to distinguish this father from other fathers of the narrator: he has only one father. The reason for presenting the content of the relative clause as an integral part of the message is not, therefore, that it expresses a distinguishing property but that it explains why the father took it for granted that the son would do as he was told.

Compare similarly:

[19]  i  He sounded like the clergyman he was.
        ii  She had two sons she could rely on for help, and hence was not unduly worried.

Both underlined clauses are bare relatives and hence necessarily integrated. But we do not understand he was in [i] as distinguishing one clergyman from another: it conveys that he was a clergyman, and an obvious reason for presenting this as an integral part of the message is that sounding like a clergyman when you are one is significantly different from sounding like a clergyman when you are not. In [ii] it could be that she had more than two sons (in which case the relative would be serving a distinguishing role), but an at least equally likely context is one where she had only two sons. In this context the property expressed in the relative clause does not distinguish these sons from other sons she has, but is an essential part of the reason for her not being unduly worried.