I'm admitted to a PhD program in mathematics, but only achieved a pass (no distinction) in masters. Will this affect my reputation in the PhD program?
I am a former 'pass' Part III student. I am going to be blunt here, as I feel emotionally attached to your case and I know why you may feel frustrated (it may be due to the typical atmosphere that part III students are supplied). Part III is just a course that people want to show-off in their CVs (only one final exam at the end of the year, busy professors, for no particular reason unhealthily competitive students). Nobody has ever asked me what my grades were in Part III. Never. I am a Ph.D., did postdocs and am finalizing a good academic job.

All that matters is your hard work in your research. Forget about the scores in Part III. If research is what you want to do in the coming years, just do it without thinking about part III results. You may end up proving that the Part III degree doesn't make any difference in anyone's life for better or worse!