"Inner child" is just a construct for a feeling you will have of being younger. If you can remember some incidents from your childhood and how you felt, when you feel the way you did then when talking about that period of your life in therapy, instead of just "remembering" how things were/seemed, that will be your inner child.

I was 52, in the kitchen making biscuits one afternoon and was thinking about therapy and about life, etc. and started having trouble with the biscuit dough; I was making them from scratch and the dough was sticky and my hands were getting all gummy and uncomfortable; I made the mental/emotional wish that my mother was there to help me contain the mess but the same instant I made the wish, I realized my mother was dead (died when I was 3) and my stepmother was, that day, in the hospital being operated on (so also very not available!). I burst into tears. That was my inner child. After a bit of time the adult, 52 year old me, was able to comfort me and deal with the sticky dough, continue making the biscuits, etc.

Transference, as hankster says, is having a feeling about a situation and misplacing the action of it onto another person/situation. You don't really know your T, that's why they do not tell us much about their personal life, so thinking she is any particular way; being loving or mean, giving or withholding, or your having difficulty asking her something like your inner child question, those feelings are often actually coming from some other, previous situation/person in your life.

One of the goals of therapy is to recognize and straighten out the feelings; kind of like I did with my inner child example above. I was not a child with messy hands needing help, I was a 52 year old adult female. In that case I was with myself and did all the work myself (which is the goal of therapy) but when a parent gets yelled at by the boss at work and then goes home and yells at the kid for some small incident, that's transference.

The parent doesn't feel they can respond to the boss so "stuffs" the feelings but a child is not going to be able to "hurt"/threaten a parent so it's safe, when the child (or spouse, or "dog") does something we don't like to respond inappropriately to that, now-occurring situation, adding some/all of the previous anger and frustration. That's what "kick the dog" refers to; we don't respond appropriately to things as they are happening in our lives but the feelings have to be resolved so we take it out on someone else on down the chain of command and it's the children and animals that are on the bottom/receiving end. A lot of the abuse and confusion we had as children was not "ours" but misplaced because our parents did not know how to respond to their own difficulties and help themselves and took it out on us.

Therapists are specially trained to be that "bottom" person for all the unresolved hurt, anger, fear, etc. and help us see, understand, and resolve it. Sometimes the feeling comes out as we are retelling the original story, as happened with my messy dough self that's the inner child. We are all the ages we have ever been.