http://revach.net/article.php?id=1512

Section: Halacha   Category: Tshuvos
Shevet HaKehosi: For What May You Daven On Shabbos?
The Halacha is that on Shabbos we do not make requests from Hashem.  However we find many parts of Shabbos davening that are requests.  For example the Yehi Ratzon after Bircas HaShachar.  What are the guidelines for Tefilos that we say and tefilos that we don't say?  The Shevet HaKehosi (3:142) says that the Maharam Mintz distinguishes between a problem that you currently have which you should not daven for (like a sick family member) and a future problem that you would like to avoid, which you are permitted to daven for.

If so he asks why does the Ran ( Rosh HaShana Perek 4) say that we don't say Avinu Malkeinu on Shabbos during Aseres Yimei Tshuva?  All the requests are to avoid troubles in the upcoming year.  The Mogen Avrohom (184:3) answers that during this period since we are in a state of judgment for the upcoming year, it is as if our problems are present before us and therefore we cannot daven for a Yeshua on Shabbos.

Disclaimer: We try to convey the Tshuva to the best of our ability. We admit that our understanding may not be accurate. Please also understand that this Tshuva may not be the final word on this topic. One should consult a Rav before drawing any conclusions.