Man, these forms are kinda outdated, someone may even blame you for using that.

Russians say: намешай бетона, насыпь сахара (сахарка, сахарку - уменьш.-ласкательная форма), отрежь сыра
"Насыпь сахарку" somehow is the only one of these which is still active.
I know well of this. I only used these pronounced forms to better illustrate my point.

Do not mix the determining noun in 1st and 2nd example!

1. Выпей чая! | Выпить [что?]--->чая! (accusative from "чай")
Выпей чаю! | Выпить [что?]--->чаю! (partitive from "чай")
чая is genetive, чай is accusative and also nominative, do not confuse the OP here. Выпить (что?) чай (accusative), выпить (чего?) чая. Чашка (чего?) чая. Подать (что?) чай. That's about it. Чаю is разделительный (partitive), which is now considered genetive.