In my experience, students don't ask for additional points in the US. If they do, I imagine they simply ask for a few "extra points". The teacher would "grant", or simply "give", these points.
Typically, students do ask for "extra credit" though, if they want to improve their grade:
Extra credit is an academic concept, particularly used in American schools. Students are offered the opportunity to undertake optional work, additional to their compulsory school work, in order to gain additional credit that would boost their grades.
Anyone can ask for extra credit. An exceptional student can ask for it to increase their grade from an A to an A+, or a failing student can request it to pass the class. I've also heard of "makeup points".
A slightly different concept you might hear is asking for a curve, if the grades aren't already curved:
In education, marking on a curve (BE) or grading on a curve (AE, CE) (also referred to as curved grading, bell curving, or using grading curves) is a method of assigning grades to the students in a class in such a way as to obtain a pre-specified distribution of these grades, such as a normal distribution (also called Gaussian distribution). The term "curve" refers to the bell curve, the graphical representation of the probability density of the normal distribution, but this method can be used to achieve any desired distribution of the grades – for example, a uniform distribution.
If the teacher wishes to do so, as mentioned above, they're grading on a curve. If the grades are already curved, then the students might ask for an easy curve. As @aschepler comments, this expression is often used to make changes to the grades that have nothing to do with a curve. For example, it could be used to give everyone a boost on a particularly difficult exam, regardless of the shape of the resulting distribution.