It is not the hosting that is the problem.
The domain owner needs to set up an SPF record to say which servers can legitimately send email for that domain, simple as that. https://postmarkapp.com/guides/spf
You can move the domain or change servers all you like, but until the person who administers the domain adds a SPF TXT record, nothing changes, as emails still leave with no SPF in place and so will more likely got flagged as SPAM.
With all due respect, you've misinterpreted the problem. The problem that the OP has is with incoming email from external domains he does not own because it is forwarded to his Zen email account from speednames.uk (the MX hosts for his domain).
Speednames.uk isn't using Sender Rewriting Scheme so emails forwarded to the Zen account have server headers which don't match the originating domain's SPF record. Zen quite correctly rejects these if the originating domain has a DMARC record requiring this - which is, in my opinion, good practice.
So, yes, it's a hosting problem. If the MX host (speednames.uk) is the same as the IMAP/POP3/Webmail host where the email is picked up from, then there's no forwarding and nothing breaks SPF/DMARC policy.