I believe Yuki's lie is, "Looks like she's in the middle of causing some more trouble." in order to convince Taka not to associate herself with Mami.
I think Taka's main lie was when she denied realising the reason Mami dropped her items for. This lie caused Mami to hastily pick up her items and breakdown. When Taka's friend came over, Mami feared that they would reveal her true intentions to Taka. Mami catches on Taka's lie and it's as if Mami's fear had come to life. At this point she realises her goddess, Taka, is not perfect. I believe this is what Taka was referring to while she and Mami were hugging.

I like linterdiction's theory about Taka having attachment issues. It has been hinted that she is drawn to those who "have something wrong with them". Perhaps Taka wants to be around those who believe that she is a goddess. Also I agree with the theory that Yuki could have also been someone who previously had "something wrong with her" and had thought of Taka as a sort of 'saviour'.