I'm writing another answer, after reading the comments, and doing some research. I am going to delete my previous answer, as I now realize it's misleading.
The measure comes from an arrangement of part of March no. 1 from "Pomp and Circumstance" by Edward Elgar. If the book doesn't mention it, it's not good.
Elgar composed the marches for an orchestra. Arrangements for piano were done by other people. In such case it's always good to check the original composition.
An example recording can be found at:
and the score at:
There are only two rhythmic layers in that section: a melody, and a steady, quarter note accompaniment. No voice plays a half note in the measure in question.
This answers the question: the C belongs to accompaniment. It is supposed to be played twice, as a quarter note each time.
The piano notation you showed is bad for several reasons:
- the curvy line is supposed to be a slur and it's supposed to refer to the top notes in the example. It should be therefore printed above the top notes
- the line is stylized as a tie. I'm not an expert of typography, and I'm not sure if strict rules exist, but after seeing many scores in my life, visually the line in the example looks more like a tie than a slur. For example Musescore draws slurs more curved than ties, and it seems right to me.
- even if we correctly interpret the slur correctly, there is one more issue: each of the C notes should be of the same length, quarter note, rather than an eight note followed by an eight-note pause.
This is a correct notation of the measure:
All this hints issues with the book you're using.
I'd like to also draw the community attention to a question I once asked on Meta: Could SE ask for more details on the scores in questions