"endeavours that" is correct and usable in written English.
You can use it to express an intention or effort to achieve something. For example: "He made great endeavours that he could pass the exam."
Both films are serious-minded endeavours that pull no punches.
But physics and chemistry Nobels often reward endeavours that began as pure curiosity.
It's the scale of my sexual endeavours that causes the problems, not the nature of them.
Follow her unusual endeavours that lead her into a stranger's apartment.
Its success or failure depends on assessing two distinct endeavours that are linked in a very unusual way.
As Ruskin said, it is not what we get but what we become by our endeavours that makes them worthwhile.
Although such principles remained the basis of America's educational endeavour, that endeavour like America itself underwent a vast evolution.
A launch of Endeavour that was scheduled for September had to be pushed back to January 2000.
There's something about a new literary endeavour that suggests tilting at windmills.
They stimulate competition in an endeavour that is often celebrated as collaborative.
Baseball is a complicated, quirky endeavour that rewards kooks who do things their own way.
