Is it permissible to add a personal request at the end of the Amidah prayer on Shabbat, such as a request for Parnasa (livelihood), a Shidduch (marriage partner) and the like?
The Talmud Yerushalmi (Shabbat, Peek 15) forbids uttering prayers for one's personal needs on Shabbat, as doing so brings to mind one's problems which can cause distress, thereby infringing upon the serenity and festive aura of Shabbat (Tur, Siman 188.) However, the Ben Ish Chai (Rabbi Yosef Chayim of Baghdad, 1833-1909) in his Sefer Torah Lishma, Sheala 103, claimed that this applies only to requests that one does not customarily recite during the week. If a person is accustomed to reciting a specific text each day requesting God's assistance in some personal matter, then he may recite that personal prayer even on Shabbat. Thus, for example, in the "Rachem" section of Birkat Ha'mazon we ask God to provide our sustenance ("Zunenu Parnesenu Ve'chalkelenu…"); although this constitutes a personal request, it may nevertheless be recited on Shabbat because it follows a standardized text. This applies as well to the "Harachaman" section of Birkat Ha'mazon, where we make many personal requests of God. Likewise, in the "Berich Shemei" prayer recited when we take the Torah from the ark, we pray for many things including children, long life and success. In all these instances, the prayer may be recited on Shabbat because it follows a formal, standard text which we recite even during the week.
Furthermore, the Ben Ish Chai ruled that even a problem that one is not accustomed to praying for may be recited to God if refraining from prayer would cause him further grief and unease. Since in this case uttering the prayer serves to alleviate, rather than cause anxiety and discomfort, Halacha would allow making the personal request on Shabbat.
Summary: One may not recite on Shabbat a prayer consisting of a personal request, unless it is a standard text that he normally recites during the week, or if reciting the prayer will help alleviate his anxiety.