What is Core Consciousness?
Core Consciousness could be described as self-awareness combined with focus and supported by very short-term (working) memory.
Notes
There is a big problem with the definition given above. It is very difficult to describe the process or experience of consciousness without using terms such as awareness, attention, thinking, thought, and so on. These terms are basically just alternative descriptions of consciousness and so we end up describing consciousness as consciousness. We just don't seem to have the ultimate ability to actually appreciate or describe what is going on in our own heads that we call consciousness.
However, it is worth exploring this a little more deeply.
Meta-awareness is, I suggest, awareness of a current state of being and context and of thoughts, intuitions, emotions, and other internal processes that relate to our current state of being.
Focus is, I suggest, a process of concentration on one or more topics or themes that is/are currently selected to be given priority attention.
Working Memory is, I suggest, the holding of a summary of the progress of attention and a collection of details related to the focus of attention.
Thoughts include, I suggest, such things as judgements, postulations, deductions, mental observations, reasonings, results, call-to-action, and demands for specific behaviour or action.
There are some additional comments that I could make about how these concepts might lead to an eventual understanding of how consciousness works but much more basic research is required.