I don't think going deeper means "catching up on" your week you had and/or spending time in chit chat. My T always cut through the chit chat, we didn't have any, that's not therapy. Therapy is what happens in the room, one-on-one with the T. That's as deep as it can get, that's all you "have" is that experience of another person and their experience of you. Everyone needs other people, there's nothing "different" or wrong about that! That's why we're in therapy, to learn to give and take and respond from the heart with other people we choose. When I started therapy my T says I was a "shell" of a person. There was no me there. My T was "there" but I could not perceive her because I was not there. That is what therapy is, learning to be one's authentic self and perceive other people. It's good that you want to have a relationship with your T. But you can only have that in that room, once a week. Everything else is "imagined" when you're outside in your regular life and wishing for your T or wondering if your T is thinking about you or what they'd say or do in a situation, etc. One can carry one's perception and feelings about T with one but that still is our own thoughts, feelings and perception, not the T's. We have to interact with the T and ask about (or be told by them sharing) their thoughts, feelings and perceptions. And that only happens in the room.

Were I you, I would have this whole conversation with your T next week. I cannot help you with your relationship with your T, only your T can help you with that and that's the whole point of therapy. Any concerns need to be talked over with the person you have the concerns with? That is what therapy is about for me; stating one's thoughts and feelings about one's T to one's T and listening to, learning to hear their response. What happened "last Thursday at work" is incidental and already passed. Only as it relates to the work being done in the room "right now" is it important.