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It is looking like I will strike out. Two of my professors, one of whom I did research for that she was very pleased with, work or worked at firms I was really interested in, but dinged me after my screeners. Is there any point in asking if they could put in a word or something, or its a lost cause?
Also, how should one go about asking a professor for help with the job search generally?

If you've been dinged, it's done with.

Otherwise, schedule a coffee/lunch and just ask for general advice. If they offer to help, great. Otherwise, get dessert, move on.