What you probably want is called a "CNAME" Record. This is like a softlink to a domain.
You would add it wherever your DNS provider has the "A record" you mention, and it would look something like
m.mainsite.org. CNAME m.my-site.org.
This tells the DNS resolver that whenever it sees m.mainsite.org, it should go try and resolve m.my-site.org
Note the trailing dots -- some DNS providers require them, some do not. When in doubt, use them.
Note that this doubles the initial lookup time for someone visiting your site; they have to do two lookups now.
A faster way that requires more maintenance is to just add an A record for m.mainsite.org that points to the IP address at my-site.org. If you do that, you will need to update both in case of a new IP address, or a failover scenario.