Weird Google indexing issues with www being forced

[Donsimong]

IM working on a site which is really not indexing as it should, I have created a sitemap.xml which I thought would fix the issue but it hasn't, what seems to be happening is the Google is making www pages canonical for some of the site and without www for the rest. the site should be without www. see images attached for a visual explanation.
when adding pages in Google search console without www some pages cannot be indexed as Google thinks the www version is canonical, and I have no idea why, there is no canonical set up at all, what I would do if I could is to add canonical tags to each page to pint to the non www version, but the CMA does not allow for canonical. not quite sure how to proceed, how to tell google that the non www version is in fact correct, I dont have any idea why its assuming www is canonical either???

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[effectdigital]

Personally it's something that I would nip in the bud, with 301 redirects. But if you are going to do that, make sure execution is flawless or you'll end up with problems

[Donsimong]

great thanks, I have access via ftp but the CMS is pretty locked down, do you think this would cause this site to perform so badly in SRPS though, its an old site and it really just does rank as I would expect. Will look into 301s thanks

[effectdigital]

You need to force one structure with 301 redirects. Canonical tags can and often are ignored, based on link and popularity metrics which Google have at their disposal. If one page has more of a legacy than its prefix variant (or more contemporary popularity) - then often Google will choose their own canonical, even if you have canonical tags deployed

Use 301 redirects to force a single structure. It can probably be achieved with one or two flexible redirect rules

Be careful of chaining. Sometimes these rules can do stuff like, do one redirect to the new protocol then ANOTHER to the new prefix. Ideally you want it airtight, all-in-one

If you are a CMS user only and your CMS is limited, stop worrying about it. There's nothing you can do