You put that in the memo field on your check.
No, the IRS does not acknowledge the receipt of an amended return. The only proof of mailing that the IRS accepts is if you mail certified (or registered) mail with return receipt or other tracking that proves delivery.
You can check the status of the amended return here, but allow 3 weeks after mailing.
I would write your SSN 1040X AND the year it's for on both the Front and Back of the check.
If lost in the mail (or lost at the IRS which happens quite often) and the check is never cashed, then without a post office tracking there is no other proof that the IRS will accept that it was ever sent at all.