"She was awarded with a medal and a trophy, and that gave here right to brag"
"She was awarded with a medal and a trophy, which gave here the right to brag"
"A recipient of medal and a trophy, she has the right to brag"
It's extremely contrived, and most will probably consider it completely ungrammatical, but I thought of:
David is more taller than Ben than than Sarah.
Which is to say, the degree by which David is taller than Ben exceeds the degree by which David is taller than Sarah.
Using the word "That" at the start of the sentence my represent two things.
1) Referring to an object mentioned previously (the simpler of the two)
For instance: That was easy! (Maybe referring to a test earlier)
2) Meaning "The fact that"
For instance: That the food was smelling bad was not in doubt.
The word "That" here can simply be replaced by "The ...
In your example, That clause is introducing a phrase which acts as the subject of a sentence. However this is not commonly used in English. The more common form is The fact that ...
The fact that the defendant believed the girl was over 13 years of age
is not defences to this charge