Does being on Clomid (clomiphene) make my period start late? This is my second cycle should I continue taking Clomid (clomiphene) if not pregnant or what should I do?

Yes and yes . Taking Clomid (clomiphene) can change your cycle. Including making it come a few days late. Just because one month doesn't work doesn't mean it won't. Usually 4 months is a reasonable course.