How do we use supposed to? In particular, should I use this while writing an official letter to ask for leave?
"Supposed to" has the same meaning as "should", but it's more intense.
Both are very similar, although "supposed to" is a bit stronger, like someone told you to go there and you are expected to come, therefore you are "supposed to go".
Sometimes you hear a phrase
Again, it's very similar to "this shouldn't have happened", although again, a bit stronger, almost like a destiny, fate.
Supposed to is formally correct, so yes, you can use it in an official document, as long as it's not misused. So how could it be misused, you might ask?
Misused example (wrong):
Obviously wrong. Unless you figured out a formula for calculating the future, you cannot make such a statement.
Perfectly proper. Peter's boss ordered him to take care of the overheating problem and Peter forgot. He was supposed to do it, but he didn't, because he forgot. You can make such a statement.