Hav! Hav!
The Zohar says that those who daven for their personal needs on Rosh Hashanah are like dogs who shout, "Hav hav — Give! Give!" From the Zohar, it seems wrong to daven for worldly needs; only daven for Moshiach and spirituality. But this isn't the accepted custom. We do daven for gashmiyos on Rosh Hashanah, and tzaddikim encourage us to do so.
One year, before tekiyas shofar, the Baal Shem Tov zy'a told his students to daven for parnassah and all their worldly needs. His students were surprised because they knew that the Zohar says we shouldn't daven for gashmiyos on Rosh Hashanah. The Baal Shem Tov explained that when Hashem grants their gashmiyus requests, it will enable them to be devoted to avodas Hashem. Thus, even asking for gashmiyos can be considered davening for ruchniyos.
Reb Matisyahu Solomon zt”l was a chazan in Gateshead. When the mispallelim came up to the words וצדקה ,תפילה ,תשובה, everyone shouted, and Reb Matisyahu zt”l shouted even louder so that his voice would be heard. His throat discharged some blood and covered the words הגזירה רוע. (This machzor with the covered words is now in London.) That year, no one died young in Gateshead.
The Baal HaTanya zt'l and other tzaddikim taught that one shouldn’t speak much on Rosh Hashanah. The Kedushas Levi takes this further and recommends that people make a taanis dibur on Rosh Hashanah. He says that when the Satan comes to speak against the Jewish nation, the court will reply, "Shah! Don't speak. It is Rosh Hashanah today, and it isn't the right time to speak and to carry on conversations. There is a taanis dibur today." Instead of carrying on conversations, we should use our time on Rosh Hashanah to daven and benefit from the specialness of tefillos said at the beginning of the year.
Praying for Material Needs
Another benefit of praying for material needs is that such a tefillah is often more sincere since these needs lie heavily on people’s hearts. The Tzemech Tzedek said that when one davens for his personal needs, it is teshuvah iloya (a very high level of teshuvah) because he is speaking to Hashem with sincerity and truth from the depths of his heart.