I think we all agree that "She is drinking hot chocolate" and "She is drinking a hot chocolate" have a slightly different meaning, the first one implying "some", the second one implying "one serving". How would you translate "She is drinking a hot chocolate" into Russian? Would "Она пьёт горячий шоколад" be incorrect?
For native English users who have a hard time distinguishing between the two:
Она sounds like uhnah Анна sounds like ahn-nuh
This is because of vowel reduction. Look this up online, it may help you distinguish words. While you're at it look up consonant devoicing (why багаж sounds like багаш) for additional help in distinguishing words.
Stress falls on the "я". I recommend using wiktionary regularly, which marks stress with an "accent" (like "é").