Pharaoh asked Moshe to pray to end the plagues in a particular way. Why didn’t he?
Various plagues were wrought by HaShem, Moshe and Ahron.  Why was barad, in particular, brought about by Moshe?
“Try and test me†Moshe replied. “At precisely what time shall I pray ×עתיר for you, your servants and your people … ridding you and your homes of the frogs so that they will only remain in the canal [i.e. the Nile]?â€
— Shemos 8:5
Moshe and Ahron left the Pharaoh. Moshe cried out ויצעק to HaShem concerning the frogs that He’d brought upon the Pharaoh
— Shemos 8:8
Moshe replied “Behold I am leaving your presence. Tomorrow I will pray  ×עתיר to HaShem, the mixed wild beasts will go away from the Pharaoh,  his servants and his people … Moshe left the Pharaoh’s presence and prayed ויעתר  to HaShem.
— Shemos 8:25,26
[The Pharaoh asked them] “pray העתירו to Hashem. There’s been too much of this Elokim-induced thunder and hail. I will send you/ your nation away; you will not have to stay.†… Moshe left the Pharaoh’s presence and exited the city. As soon as he spread his palms up ויפרוש כפיו to HaShem the thunder and din ceased and the hail and rain no longer fell to the ground.
— Shemos 9:28,33
There are six things which HaShem hates, seven which His Soul abominates: 1. stuck-up eyes, 2. a lying tongue, 3. and hands that shed innocent blood; 4. A heart that works out malicious thoughts, 5. feet that are quick in running to evil; 6. A false witness who exhales lies, 7. and one who causes conflict among brothers.
— Mishlei 6:16-19
Rabbi Chanina the son of Dosa would say … One whose deeds surpass his wisdom, his wisdom endures. But one whose wisdom surpasses his deeds, his wisdom does not endure.
— Pirkei Avos 3:9
There are 10 different expressions [in Lashon Kodesh-the holy tongue;] for prayer …
— Sifri on Devarim 3:23
In an abstract way we are aware of the Chazal that teaches that there are 10 near-synonymous expressions in Lashon Kodesh to describe humans communicating with HaShem. On a theoretical level we are also cognizant of the fact that diverse words carry assorted shades of meaning and that, as such, there must be 10 different ways to pray, 10 distinct media for prayer.
Yet, we are accustomed to congregational prayer during which everyone must be on the same page, both figuratively and literally. We also pray using a liturgy fixed by the anshei k’nesses hagedolah-the men of the great assembly; with later accretions canonized by tradition. And so on a practical level for us there is only one way to pray. Gradations in the quality of our prayer vary according to levels of ones understanding of the liturgy and ones sincerity and depth of kavvanah-directing his heart and attention towards G-d. To us, the notion that varying circumstances require a different substance or even style of prayer seems utterly foreign.
In Parshas VaEra the Izhbitzer school teaches that the style and substance of prayer must react and respond to the particular needs being addressed and to the root causes of the distress that one is praying to resolve. Just as no two crises are exactly alike so too no two prayers can be clones of one another.
In each of the makkos-plagues; of frogs, mixed wild-beasts and hail we find the Pharaoh of Egypt beseeching Moshe to pray for the cessation of the makkah. The Pharaoh is consistent. Every time he requests Divine intercession of Moshe he employs a conjugation of the word עתירה atirah-pleading. Yet only in requesting the end of the makkah of the arov– mixed wild-beasts; does Moshe actually plead with HaShem. In order to get the frogs back into the Nile Moshe employs tzeakah-shouting or screaming;  and to stop the makkah of barad-hail composed of fire and ice; Moshe prays with perishas kapayim-spreading his palms outwards and upwards. The second Izhbitzer Rebbe, the Bais Yaakov, offers insight into the three crises and why the three different prayers were appropriate for each one.
Observing that both the makkos of tzefardea-frogs; and arov were incursions of wild animals into human habitats, the Bais Yaakov asserts that all creatures, both domesticated and wild, yearn for the proximity of human beings for they have a deep-seated, instinctive consciousness that their own actualization and fulfillment can only be brought about by human beings. But for the vast majority of baalei chaim-animals; hobnobbing with human beings is not the proper means through which man might perfect and fulfill them. Among the Creator’s creatures Man alone is endowed with free-will and thus, with the capacity to exercise free-will to serve G-d. These acts of avodah-serving HaShem; distinguish man from beast and are what drive away undomesticated animals from human habitats. The power inherent in various types of avodah is what make the different baalei chaim maintain their distance.
The croaking frogs and toads are distinguished by their ability to give voice to wordless cries, groans and screams. They have voices, but their voices cannot inform words. Correspondingly, the type of prayer-based avodah that keeps frogs separate and distinct from human society is human tzeakah which is similarly inarticulate and wordless. When tzeakah is wielded by a human being it is a non-verbal, yet voice-based, form of communication. This is why, when the time came to end the makkah of tzefardea, Moshe prayed with tzeakah.