Q. Can one daven on Rosh Hashana that coincides with Shabbos for one’s personal needs such as health and parnassa (sustent)?

A. Shulchan Aruch (O.H. 288: 9) permits in cases of danger of life to pray for those needs on Shabbos.
Mishna Berura (28) thus explains why on a regular ‘Mi Sheberach’ we add the words; ‘Shabbos Hi Milizok’ oe we should not be praying on Shabbos.

Poskim point out that there seems to be many contradictions regarding the above Halacha, since we do recite in Shabbos the Yehi Ratzon after the morning blessings, E-lokai Netzor at the end of the Amida, Yekum Purkan and the Tefilos of Birchas Hachodesh, just to mention a few.

Medresh Tanchuma (Vayero) mentions that the reason for avoiding those Tefilos is to spare ourselves from the suffering and pain involved when reciting them. Therefore, if they are standard one should be able to recite them. Yet we do daven a short Amida on Shabbos. (See Piskei Teshuvos ibid).

As far as Rosh Hashono is concerned, Remoh (584: 1) rules that we do not say Avinu Malkenu in Rosh Hashana that coincides with Shabbos for the above reason.
Horav Shlomo Miller’s Shlit”a opinion is that you may daven for the needs of others.

Rabbi A. Bartfeld as revised by, Horav Dovid Pam, Horav Aharon Miller, Horav Chanoch Ehrentreu and Horav Kalman Ochs Shlit’a