"Tá glasra ag an bhfear."
Translation:The man has a vegetable.
Since I'm doing this lesson on my mobile phone and thus don't have access to the grammar notes until later, when I can check the website, could I just check my understanding here with someone more knowledgeable? Is it the case that all/most [preposition] + "an" phrases cause eclipsis of the following noun unless it starts with a vowel? Does that happen with both masculine and feminine nouns?
Yes. Most do (don does not), and, yes, it happens on both genders. Note, however, that in some dialects it lenites instead of eclipses