He pretended not to see me to avoid an awkward situation.
This sentences uses negative infinitive. Negative infinitives are made by putting not before the to+infinitive.
After many verbs you can use infinitives as well as negative infinitives. That sentence means:
He pretended that he didn't see me to avoid an awkward situation.
Your second sentence is not grammatical because it lacks a subject. In other words, that is a conjunction which relates two clauses. You should put a subject after that.
Note: This answer is related to the original question not to the edited version.