Reference: Fataawa Noor ‘alad-Darb – Question No.6309, Page 142, Volume 12
Question: Is it true that the supplication does not ascend for acceptance to the heavens unless it is preceded by prayers [and salutations] upon the Prophet ﷺ? And what is the evidence for that?
Response: [Extending] prayers [and salutations] upon the Prophet ﷺ before the supplication, as well as beginning with praising Allaah is best, and something which has been legislated. So if anyone wanted to supplicate to Allaah (‘Azza wa Jall), then he should begin with praising Allaah, then [extending] prayers [and salutations] upon the Prophet ﷺ – and then supplicate [to Allaah] as he wishes.
And if he were to supplicate [to Allaah] without [beginning like] that, then there is no harm in doing so. If he did not start with praising Allaah (Ta’aala) or [extending] prayers [and salutations] upon His messenger ﷺ – then this will not [necessarily] prevent his supplication from being accepted.
As for the hadeeth which suggests this [to be the case], then it is not a narration that comes from the Prophet ﷺ:
…however, I cannot at this moment recall the extent of the authenticity of this hadeeth.
 Transmitted by at-Tirmidhee No.486 – with a mawqoof isnaad – where the chain of narration ends at ‘Umar ibn al-Khattaab (radhi-yAllaahu ‘anhu), and not the Prophet ﷺ.
Ibn Katheer said: Its isnaad is good; and al-Albaanee classed this narration as hasan in Saheeh at-Tirmidhee.
al-Albaanee also mentions in as-Silsilah as-Saheehah No.2035: ‘In conclusion, when all these chains of narration and corroborating reports are taken into account, the hadeeth is hasan at the very least, in shaa.-Allaah.’