Step 1 Create /etc/mail/authinfo with one line (terminate it with a line break):
AuthInfo:mysmtp.com "U:root" "I:<mysmtp.com login name>" "P:<mysmtp.com password>"
Step 2 Run makemap of this file to create a authinfo.db-file
makemap hash authinfo < authinfo
Step 3Correct your sendmail.mc-file (whatever it is called on your system). Add these lines:
define(`SMART_HOST',`[mysmtp.com]')dnl
define(`RELAY_MAILER_ARGS', `TCP $h 587')dnl
FEATURE(`authinfo',`hash /etc/mail/authinfo')dnl
Note: You should change this for the sendmail queue runner process, not a SMTP submit service etc.
Step 4 Compile your sendmail.mc file into a sendmail.cf file.
make cf
Step 5 Restart sendmail.
make restart
Also: remember to correct your SPF record and similar setups on your domains according to the setup e-mail from mysmtp.com