Using CNAME with MX record isntead of Nameservers?

I’m migrating a client’s website to a different hosting provider.
This provider uses an A record and CNAME. My client also has an MX record pointing to a mail server, this looks like this: mail.mycompany.example

Long story short I can’t transfer the domain from the old web company to the client yet.
Their old web company added the CNAME and A record to the DNS entries, the new site was live but the client’s email went down, my guess something with the MX record.

My question is there an incompatibility with using a CNAME and MX record? or is this company being petty?

Is this setup valid? Type | Name | Value

A Record | @ |  XX.XXX.XX.XXX
CNAME: | www | company.hosting.example
MX Record | A |mail.mycompany.example