I've read evidence for both sides:
NO, you should not create a separate mobile navigation.
Research by Sujan Shrestha (see page 189) showed that omitting part of the desktop navigation on mobile can be disorienting to users who are familiar with the desktop site, reducing their task completion times, so you're right to think twice before changing the navigation for mobile.
Also, by hiding some of the pages from mobile, you're implicitly assuming a separate mobile context. Roughly 70% of mobile device access occurs in a stationary context (Heimonen found 67% in 2009; Church & Oliver found 71.4% in 2011, p. 71). People in these stationary contexts (and potentially even mobile users in mobile contexts) will have the same information needs as desktop users, so hiding the relevant part of your navigation will make it difficult for them to find the information they seek.
YES, you should create a separate mobile navigation
The authors of “Improving Web Search on Small Screen Devices” found that users of mobile devices accessing desktop sites often failed to complete tasks on desktop-optimized websites:
"the main reason for failure, and the associated large task timings, was the great difficulties they had in navigating the site selected from the search result. Most of their wasted time and effort was spent in becoming
increasingly lost within the small window" (488)
Unless the content is essential, by providing links to it, you'll only cause your users frustration as they struggle to use an oversized interface in an undersized screen. By removing that part of the navigation, you'll improve the experience for all users who are just browsing casually and don't really need to complete tasks on those pages.
A caveat: some other researchers found that (at least in the context of mobile apps) browsing accounted for only 10-12% of the total usage (p. 186), which means that users will often be using the internet with a purpose in mind. Even if that purpose is just finding information or checking a status, you will prevent them from doing so if you hide the pages on mobile. For this reason, if I were to pick one side personally, I would choose not to hide the desktop pages from the mobile navigation. Being frustrated by bad access is not as bad as being frustrated by having no access at all.
NEITHER (the unrealistic ideal)
Ideally, the navigation for the two should eventually be the same because ideally the content should be the same, with both desktop and mobile sites pulling their content from the same database, but just formatting it differently. In her book Content Strategy for Mobile, Karen McGrane shows how cross-platform content publishing should work. It sounds like your company's forked their content for the time being, so this probably isn't a viable solution at the moment, but it's something to keep in mind for the future.
A WARNING. If you do choose to hide pages from the navigation...
...then consider a mobile navigation pitfall that McGrane warns about:
Mobile users entering your site from Google could get redirected to the mobile homepage if the page doesn't exist on mobile, then have to go to the desktop version of the homepage, and then have to search laboriously through the desktop version's internal navigation to find the page they're looking for.
If a user can find a page on Google, they should be able to access it on mobile, even if it's only the desktop version of the site. If you decide to omit the content from the header and footer, be sure that you do not also redirect search results from Google that would have gone to those pages.
One last note...McGrane also recommends looking at competitor's sites to help determine your mobile strategy:
how do they handle global navigation? Does it include the same
major categories as the desktop site, or just a subset? Are
navigation categories prioritized differently for mobile? How does
the user access the global navigation from the homepage and other site
pages?
...so doing a competitive audit might help you to answer your question.
