Learn Russian: Каждый вечер мы пили чашку чаю на веранде. - Every evening we drank a cup of tea on the veranda.

This relates to our current lesson on repeated past sequences. Even though finishing a cup of tea is a complete result, the trigger Каждый вечер makes the overall situation a recurring habit. Habitual repetition in the past overrides the completion of the individual action, so you must use the imperfective пили.

Why is the word for tea чаю instead of the standard genitive чая?

This is a special form called the partitive genitive, sometimes called the second genitive. It is used for uncountable mass nouns when talking about a specific quantity or a portion of something (like a cup of tea). For чай, the -ю ending is very traditional and natural here, though the regular genitive чая is also acceptable.

Could I use the perfective выпили to show that we completely finished the cup each time?

In a simple standalone sentence with a frequency word like Каждый вечер, no. Standard Russian requires the imperfective for simple habitual past actions. The rule is that repeated past actions must use the imperfective, even if a specific result (a finished cup) was achieved each single time.

Why do we use the preposition на for the veranda instead of в?