Syahir very kindly responded to my support query as dialogue quoted below.
I would therefore propose this facility as an enhancement.
" When using graphene mobile, how can I display a header image at the top of each page?"
A from Syahir
" Unfortunately there is no support for custom header image in Graphene Mobile yet. This is mostly due to the variety of dimensions of different devices and orientation, that implementing such a feature is not a straightforward process.
For the time being, if you're handy with coding, you can probably implement a temporary workaround for the feature yourself using pseudo-child theme for Graphene Mobile. "