Kvod Horav
Q. Is it better to pray for the sick at the refaenu bracha or during shomea tefilah?

A. Shulchan Aruch (O.H. 119: 1) maintains that a person is allowed to add to any of the middle brochos of the amida, a prayer of his own that is similar or in the context of the theme of this particular brocho, such as davening for someone sick on the brocho of refoenu etc.
On shomea tefila one may daven for all his necessities, since this brocho includes everything one may need. Mishna Berura (ibid. 1) adds that on refoenu one should only supplicate for someone that is already ill, and not to avoid future sickness or keeping health, that should be delegated to shomea tefila.
Ran (Avoda Zara 8a) explains that if one is praying for an existing patient, he should do so immediately at refoenu and not wait until shomea tefila. Shaarei Teshuva (116: 1) quotes Hari Molko that it is better to pray expressing words on shomea tefila and during refoenu he should engage only in beseeching thoughts. He then mentions that others maintain that it is better to pray words immediately at refoenu.
Ohr Letzion (2: 7: 33) deduces from the expression of the Talmud, that if one has an ill person at home, it is preferable to ask for him at refoenu, however, if he is pleading for someone not close, he should do so at shomea tefilo.
Shevet Halevy (8: 21) quotes Sefer Chasidim that it is proper to add personal tefilos and petitions to all middle brochos, since it creates devotion and intent to a tefila, that may otherwise be said without much kavanah.
Horav Shlomo Miller’s Shlit’a opinion is that when the added tefilos are constantly said on every amida, it is best to recite them at the end of shemone eserei on Elokai netzor, since their everyday recitation may seem to turn this additions into an integral part of the brocho. The Rov added that the prior argument that the additions create devotion and intent, diminishes when they are constantly repeated and become banal and timeworn. (Aruch Hashulchan ibid. rules similarly)

Rabbi A. Bartfeld as advised by Horav Shlomo Miller Shlit’a