moikkai (もいっかい): one more round

This means one more round, one more rotation. The kanji for kai is 回.

otooshi (おとうし): appetizer served with alcohol

Otooshi, also known as sakizuke and tsukidashi, is a side dish eaten with alcohol that is automatically served before a meal at an izakaya (Japanese style bars), a sushi bar, and a traditional Japanese restaurant regardless of your order.

Reference: Gourmet Navigator

hitohada (ひとはだ): body temperature

hito is person and hada is a person's skin (or outer surface). This is used to describe a temperature and Ayumu used this at Cosmos to describe the temperature of a drink.

nomitarinai (のみたりない): not drinking enough

Tariru is like sugiru, it is an "add on" verb and it means to be enough, to suffice. Tarinai is the negative form, so nomitarinai is used to say that someone isn't drinking enough.