to invoke blessings upon the Prophet ﷺ was left unqualified by the author—may Allah have mercy on him—so the scholars, may Allah have mercy on them, differed: Is sending blessings upon the Prophet ﷺ obligatory once in a lifetime, obligatory only on certain occasions, or not obligatory at all? The sound position is that it becomes obligatory whenever one of those occasions arises; otherwise, in its original ruling, it is recommended.
What, then, is the meaning of invoking blessings upon the Prophet ﷺ? In other words, what does the supplicant mean when he says, “Allahumma ṣalli ʿalā Muḥammad (O Allah, send blessings upon Muhammad)”? Most people recite or utter this supplication without knowing its meaning, and this is a mistake. Whatever you say, you should know what it means; whatever you ask for, you should understand, so that you do not unwittingly pray for something sinful.
Your statement “Allahumma ṣalli ʿalā Muḥammad” means: “O Allah, praise him in the Highest Assembly.” To “praise him” is to mention him with noble qualities, and the Highest Assembly refers to the angels. Thus, when you say “Allahumma ṣalli ʿalā Muḥammad,” it is as though you were saying, “My Lord, describe him with praiseworthy attributes and mention him before the angels so that their love for him increases and they are further rewarded for it.” This is the meaning of “Allahumma ṣalli ʿalā Muḥammad.”
The scholars—may Allah have mercy on them—also differed on whether it is permissible to invoke blessings upon someone other than the Prophet. That is, may a person say, “Allahumma ṣalli ʿalā so-and-so, the scholar,” or “Allahumma ṣalli ʿalā my father,” and the like? The correct view contains some detail. If the prayer for others comes as an adjunct to the prayer upon the Prophet ﷺ, there is no harm; for this reason, when the Companions asked the Messenger of Allah ﷺ how they should invoke blessings upon him, he said, “Say: Allahumma ṣalli ʿalā Muḥammad wa ʿalā āl Muḥammad (O Allah, send blessings upon Muhammad and upon the family of Muhammad).” If the invocation for someone else is made independently, then provided there is a legitimate reason, there is likewise no objection.