The short answer is, yes of course there's hope for you! Most people (barring actual physical impediments) can learn to sing in tune.
BUT (and it's a big but) learning to sing is a bit like learning to drive. Some people "get" it almost immediately - 3 lessons and then they pass their test first time. Others spend ages - 20,30,40 lessons - and fail the test umpteen times before they get through. But they get it in the end. And their ability in the end may be no less than their faster learning counterparts.
You may be one of those who need to plug away at it for a long time before it comes. However, frequently people who don't "get" learning to sing fairly quickly just give it up - it's not quite such a must-have skill as driving when it comes to such things as job applications - which is in part where the "either you can do it or you can't" myth about singing comes from.
So what you need to think now is whether you want to invest the time and effort necessary for what is really a very small part of a test that is in itself a small part of the whole exam. Given that the exam is imminent, it may be better to spend that time putting an extra polish on your pieces to gain a mark or two there instead.
In the meantime a few tips for the exam itself.
See if humming or whistling improves your ability to pitch the notes.
You will get some points simply for attempting the section - only a total refusal gets a zero score. And you will get more points for getting the rhythm correct. So I would suggest - at least for your next exam - to concentrate on getting the rhythm right. Go up and down according to the music but relax about the pitch - give yourself permission to get it wrong.
And don't neglect the other parts of the aural test - you can make up more points by being sound on them.
Finally - Good luck with your exam!!