Pip and Miss Havisham have a complicated relationship. She relies on him because she needs someone for Estella to practice on. Although she hates all men, and presumably boys, she does seem to take a subtle interest in his life. He continues to keep in touch with her as he gets older, and she allows him to and even continues to push Estella his way. She tempts and teases him, but she never tells him the truth. She allows him to assume that she is his benefactor.
Pip is irritated by Miss Havisham, and even confronts her after he realizes that Magwitch is his benefactor. Even then, he realizes that his life is as bad as she could have hoped.
“What I had to say to Estella, Miss Havisham, I will say before you, presently—in a few moments. It will not surprise you, it will not displease you. I am as unhappy as you can ever have meant me to be.” (ch 44, p. 242)
Miss Havisham asks him why he is surprised by her unkindness, and she has a point. She has been nothing but manipulative throughout his life, and he was always aware of it. He put up with it first because he thought he had no choice, and second because he thought he was going to get Estella.
Pip does care for Miss Havisham, because in a twisted way she is family. He feels sorry for her, at the end, and jumps into the fire to rescue her.
Miss Havisham finds a companion in Pip. She is a lonely woman who has had a lot of time to brood over her past and has a lot of selfishness and bitterness that she at first takes out on Pip because she realises he is vulnerable and looking for answers.
She sees Pip as just another man on which revenge should be taken, due to her anger and humiliation after a past love affair with Compeyson. At the same time she finds him to be a wonder to her and yet fears him.
Pip sees her as his benefactor and despite her plots and the harm she causes him, he never rebukes her because he expects great wealth from her and expects to be trained to be a gentlemen.
The best way to describe their relationship is: volatile , "rocky", unstable.