welcome to WebmasterWorld [webmasterworld.com], mywebguytaylor!

my clients is splitting their backlinks from example.com and example.com/home

by "backlinks" i assume you mean these are external links.

which of these two is the canonical url?
(hopefully it is https://example.com/ or https://www.example.com/ )
are the requests for these urls being redirected (with a 301 status code) to the canonical url?

These are links pointing from other websites into these two urls on the website...

which of these two is the canonical url?
(hopefully it is https://example.com/ or https://www.example.com/ )
are the requests for these urls being redirected (with a 301 status code) to the canonical url?

Are example.com and example.com/home the same page? Can't help thinking that, if so, your client's site is due for a cleanup. Preferably before the search engines start yattering about Duplicate Content.

Are example.com and example.com/home the same page?
YES

example.com/home should be redirected with a 301 status code to example.com/