You can, for example, assign each menu sub-item its own template style. This means that the header image is on one menu item and not on the other.
For this you go in the Joomla backend under SYSTEM -> Templates - Site Template Styles > and there are already about 12-14 ready-made styles that are assigned to certain pages of the website. You can then rebuild it or create other styles. in the respective style you can then select the pages on which you want to display it under "MENU ASSOCIATION"...

I must admit I'm struggling with this. I can create a new template style, e.g. same as in use but with no headers, I notice there is a tab "Menu Assignment" which i can seemingly select which menu items this style will apply to. Having done that I save and look back at the list of styles. The previous one which had header images selected is gone and replaced on all pages with this new style with headers absent on all pages. Presumably because the template style is assigned as "Default" and is confirmed under the Pages column as "Default for all pages". I see no way of turning this off and adding a second style to this list.
I tried loading the preset styles (6 of them and they don't vary much - just a few changes to some colours) these also all end up default for all pages.

It seems logical that using the Menu Assignment tab would define which pages the style works on but it doesn't seem to do anything. I notice that Magnolia is one of your older templates. Is this a factor?

Pleased to say I have solved the problem.

I missed the drop down option on saving the template style.

It is necessary to drop down and Save as Copy in order to get a second template style.

So used to quickly clicking on save in Joomla I didn't notice what was right in front of me.