With the door open, shove a screwdriver into that slot, as if it were the door striker on the pillar. As you slide it along the slot, the claw should go to a "first position" (like the door not being properly shut) and then the "second position" (door fully shut). Once that has happened, pull either the interior or the exterior handle and the claw should spring back to the "door open" position. If it doesn't, you might need to help it with the screwdriver while someone works the door handle.
Sadly, the door latch assembly is usually a non-serviceable unit, so it would need replacing. However, there's no harm in trying some spray lube (ideally spray grease) down the side of the claw, towards wherever it looks like the pivot point(s) might be. You could also take the door card off and see if the back of the latch assembly is any more accessible inside the door?