By HaRav Shimon Anshin shlit”a
A. When a serious incident happens and a person needs to pray for Mercy:
In such situations, the Halacha dictates that a person can prayer for Mercy, even to the degree of “prostrating oneself…“ (Magen Avraham 288:13, Misnah Brurah 26). In fact the Shulchan Aruch Harav (Baal HaTanya) writes that in such cases a person should prayer for Mercy immediately!
B. Requesting one’s future needs:
There are those that permit requesting one’s future needs on Shabbos, meaning that one should only prayer for ones future needs and NOT for one’s current needs on Shabbos (Questions and Answers, Halachos Katanos part I 62). However, those that prohibit the above approach, say that on the contrary, praying for one’s future needs on Shabbos is even worse than praying for one’s present needs (Questions and Answers Yaabetz 64).
D. Praying for one’s Spiritual needs:
According to the language of the Mahara”m Mintz (siman 87), the reason that one is permitted to prayer “Elokay Ad Shlo Notzarti …” (At the end of the Shemonah Esrei dealing with one’s spiritual needs) is because it is not considered a request for one’s needs with the exception of asking for Parnasah etc, which could make it appear as if a person is in a state of grief or anguish over his physical needs.
However, showing regret/remorse for one’s sins is good to say every moment each and every day.
Similarly, it is brought down in Questions and Answers of the Baal HaTanya, that it not forbidden to beg for one’s needs again with the exception of Parnasah etc i.e. those things that are bodily/physical needs. However, the Baal HaTanya also mentions that showing regret for one’s sins is good to say everyday.
In summary then, the difference between these prayers boil down to praying for one’s physical needs which is forbidden and praying for one’s spiritual needs which is permitted.
E. Confessing one’s sins:
According to the Mahara”m Mintz it would appear as if this is permissible as it is not considered “requesting one’s needs” per say, however, the Mishna Berurah prohibits such confession.
F. Crying on Shabbos:
Crying out of emotional feeling towards Hashem, is permitted according to all Poskim. Even though, there are those that prohibit crying “to release the sorrow from one’s heart”, in practice it is permissible (Shulcha Aruch 88:2, Mishna Berurah siman katan 4).
G. Making a “Mishebeirach”:
It is permissible to make a “Mishebeirach” for an ill person who is in a life-threatening situation, and one is allowed to prayer for such an individual in any form or way. For example as formulated in the Siddur: “…lachlimo, v’larapaso …” (“for his recovery and healing ….”)
In the case of a non life-threatening sickness however, most Poskim forbid making a “Mishebeirach” in the way of a formal prayer. Many have the incorrect Minhag of using this formulated “Mishebeirach” prayer specifically for those that do NOT have a life-threatening sickness or, they include the name of such an individual together with the name of those that do have life-threatening illnesses (Heaven Forbid).
Therefore, the correct Minhag would be to recite a special “Mishebeirach” for the non-seriously ill as is found in many Siddurim, which concludes in the way of a “Brocha”: “V’Bischar zeh, HaKadosh Baruch Hu Yerachem Alav …” (“And with this merit (of giving Tzedakkah, Hashem should have Mercy on Him …. “) .
As this is a translation of the original Hebrew, if you are unclear on any of the Laws outlined herein in any way whatsoever, please consult with a Posek (Halachic Authority).