The answer for this is somewhat simpler than one would expect, but it does complicate a lot of other stuff regarding this time period. Simply put: The Seljuks never received such a letter. Its one of the most successful pieces of propaganda ever!

The earliest sources known to have written about the Seljuks in detail is the Malik-Nama and some notes from an Armenian monk called Bar Habeaus. Unfortunately, the Malik-Nama is lost to us and we're aware of its existence only because a handful of later texts reference the Malik-Nama as the definitive word on Seljuk history. In contrast to all the later work on the Seljuks, the works of Bar Habeaus and every work which references the Malik-Nama make literally no references to the Caliph ever begging for help from the Seljuks! They do mention a letter but the tone of this letter is much more belligerent than what we're asked to believe by later historians. The Caliph in this letter chastises the Seljuks and demands they stop killing innocent Muslims and send tribute immedeately - shockingly, Tughril beg, who is painted in later sources as a pious and fair Muslim essentially sends back a letter in which he says that he'll happily kill those who stand up to him and that he'll send tribute if he can! His co-regent and brother Chagri Beg refused to even sign the letter because he allegedly had such contempt for the Caliph.

I mean, Chagri and Tughril were the first Seljuks to take their people out of the shamanistic steppes and into the dar-i-Islam so their apathy is actually not particularly shocking, what is shocking is that later Seljuk authors did such an incredible job of hiding all this. Remember that all of the historical works which reference the letter begging for help and the piety of early Seljuks were written during the height of the Seljuk period by scholars literally paid by the Seljuk government!

To further prove the letter to be a fabrication, merely examine some of the other discrepancies between early and later sources. In the earlier Malik-Nama associated sources, it is written that that Daqha, grandfather to Tughril and Chagri, fought a war with some Khazars because they stole some of his camels and sheep - a common occurence on the steppe. In later sources, it was to prevent the Khazars from attacking Muslims. In later sources it is alleged that Seljuk talked the Khazar leader out of raiding Muslim cities and then distributed Qurans among his tribesmen. In reality Seljuk was almost certainly illiterate and the early sources make no reference to his arguments with the Khazars.

So essentially the sources which allege the existence of this letter all seem to be full of fabrications which clash not just with what we'd consider plausible but with all of the contemporary evidence. In reality Tughril and Chagri had extremely poor relations with the Caliph and the sedentary Islamic world - especially Chagri who had an enormous fight with his brother over being allowed to sack Nishapur during Ramadan, when he allegedly held a dagger to his own throat and told Tughril that he'd kill himself if he weren't able to sack Nishapur. Eventually Tughril said that it'd be impious to sack the town during Ramadan and told his brother he'd have to wait till 'Eid to slaughter the men women and children inside... how generous of him. That really tipifies the lip service paid to the church by the Seljuks who had barely been out of the steppes a few decades by the height of their empire.

Secondary Sources:

The Poltics of Knowledge in Premodern Islam (Omid Safi)

Islamic Civilisation 950-1150