How to send mail from a dedicated server if the sender (postal address) is delegated to another mail service?

Question

Mail server

Leaving mail messages from the server using the mail() function.
The sender is [email protected], the recipient is anyone, and the blind copy is my mail, respectively. So, the mail of the xxx.com domain is delegated to the mail.ru service, I send a letter, it comes to my mail (blind copy), but it never arrives, no matter what the sender is, to the mail addresses of the xxx.com domain.
The system administrator says that it is impossible to configure this, and you need to do all this through the smtp server of mail.ru itself, but they have a limitation, and in this case messages can go quite often, so for what reason does it not come there?
Anticipating one of the issues in the domain settings are specified:
txt:
v=spf1 a mx ip4:xxx.xxx.xxx.xxx/32 include:_spf.mail.ru

Answer 1

Raised something like this, except that the mail was delegated to Yandex.
Through the same smtp it sends calmly, except that, in the DNS at the registrar, he prescribed a DKIM digital signature so that the mailers do not swear at the recipients.
I don’t know what restrictions the mail has, I even carried out mailings through Yandex smtp, through which, in the same Yandex, one could also look at the statistics.