What is the meaning behind the scene where Saki fights a Tainted Cat?
I think Saki used her skill - restoring object. She was seen training earlier fixing vases and stuff in school. This ability allowed her to fix the broken necklace (for a while), during the attack and heal herself.

I believe that the point was to strengthen the reference that Saki has the same (unique) ability as Tomiko Asahina and is capable restoring own cells and even their telomeres. This is only made clear later by Tomiko Asahina herself, saying that she plans to make Saki her heir, somewhat base on her ability.