Can I ask a professor to re-mark my essay?
You can ask, and at many institutions the request has a formal route, but frame it as a question about the mark against the published criteria, not a request for more marks.
You can ask, and at many institutions the request has a formal route, but frame it as a question about the mark against the published criteria, not a request for more marks. Ask whether a re-mark is open for that assessment before writing: some assessments are closed to re-mark requests when they were marked independently by two markers. If the route is open, expect to submit the work and a written explanation of which criterion was applied incorrectly.
The formal version usually comes with its own paperwork: a request form, the marked copy with the comments, and a written statement that names the criterion you believe was misapplied. At the second stage the work can go to an independent reader who is not told the original mark and scores it fresh, and the outcome replaces the mark whether it is higher or lower. Keep in mind what is actually being changed: a re-mark changes one assessment mark, and the course grade is then recalculated from it, so a two-mark gain on a twenty-percent essay rarely moves the final grade far.