To set up email for a domain name you generally need to:
- Set the DNS MX records to the host name of the mail servers
- Set subdomains such as "mail", "smtp", "webmail", or "mailbox" as CNAME to the mail host so that checking mail works.
This does not usually interfere with web hosting. Web hosting usually has a DNS record for the naked domain as well as for subdomains such as "www" and "blog". In some cases there may be a wild card entry for all subdomains.
I've never had a mail host ask me to set CNAME entries that aren't generally reserved for mail use or that interfere with web hosting. In fact, looking at the instructions for setting up Zoho, they want a CNAME for the "mail" subdomain. I would be very surprised if you are hosting any web content an a subdomain called "mail". Your domain can have multiple CNAME entries, one for each subdomain.