Why is saki in japanese written as さつき？ there's other times I've seen the tsu in words without being pronounced. Why is this?
You're probably be confusing つ and っ. In other words, it's not さつき you're seeing, but さっき.
The small っ is not, however, silent: it creates a slight pause between さ and き, meaning words like さっき and さき, or 活気 (かっき) and 下記 (かき) are not homophones.