Q. Would it make a difference if that day coincides with Shabbath?
A. As mentioned on question above in principle one should avoid davening for personal needs during Shabbos and indeed therefore we don’t recite Avinu Malkenu when Rosh Hashana coincides with Shabbos.
See Piskei Teshuvos (287: 1) that it is permitted when the request is recited as a blessing such as Mi Sheberach or Yokum Purkon. See Remoh 284: 7.
Rabbi A. Bartfeld as revised by Horav Dovid Pam, Horav Aharon Miller and Horav Chanoch Ehrentreu Shlit’a.