Unless hyperlinked in the text, the sources cited below can be found here.
1) Yesterday I purchased the "hot off the presses" Yabia Omer - volume 11, by R. Ovadia Yosef. Though published after his death, all of its teshuvot were written by Hakham Ovadia at various points during his lifetime.
Flipping through the book, I was instantly drawn to סימן כג. Written in 1951, it is a response to "רב אחד," who had criticized Hakham Ovadia for what he perceived as Hakham Ovadia's "harsh condemnations" of R. Yosef Hayim - "Ben Ish Hai," in his first book, Hazon Ovadia.
(Regarding R. Nissim Kadouri's involvement in this critique, as mentioned at the beginning of the teshuvah - see R. Yisshak Yosef's related memories here).
(Regarding R. Nissim Kadouri's involvement in this critique, as mentioned at the beginning of the teshuvah - see R. Yisshak Yosef's related memories here).
2) The first specific critique related to Hakham Ovadia's pesak regarding the recitation of "לשם יחוד" prior to prayers and performance of missvot. Whereas Ben Ish Hai advocated for the recitation of these kabbalistic introductory prayers and even composed several himself, Hakham Ovadia brought support for the well-known opinion of R. Yehezkel Landau (see his bio here), who argued in his Noda BeYehudah that the recitation is unnecessary for the proper fulfillment of prayers and missvot. In this particular teshuvah, Hakham Ovadia defended both his pesak and particular wording - proving that he had not intended any offense to the authority and stature of Ben Ish Hai.
3) The historical backdrop to the pesak of Noda BeYehudah is quite interesting. As R. Yisshak Yosef explained in a shiur last year, R. Landau served as the rabbi of Prague during the middle of the 18th Century, and therefore feared the threats of the prominent movements of Sabbateanism (followers of the false-messiah Shabbetai Sevi) and Hasidut - which were pronounced by a strong dedication to mysticism and kabbalah.
In her The Kabbalistic Culture of Eighteenth-Century Prague: Ezekiel Landau and His Contemporaries, Prof. Sharon Flatto explained that although R. Landau was both accepting of kabbalistic thought and well-versed in its practice, the threat of Sabbateanism and Hasidut compelled his strong opposition to the recitation of לשם יחוד.
The most famous line in Noda BeYehudah's teshuvah is:
ועל הדור היתום הזה אני אומר: ישרים דרכי ה' וצדיקים ילכו בם וחסידים יכשלו בם
He likened his "orphaned generation" to the realization of a verse in Hoshea. Whereas the verse actually contrasts the proper ways of God and the righteous to those of the wicked ("פושעים"), R. Landau replaced the "פושעים - wicked" with "חסידים - righteous" - hinting at his aversion to the "new" forms and movements of "hasidut - righteousness" in his generation.
4) R. Landau's student, R. Eleazer Fleckeles (see his bio here) related in his Teshuvah MeAhavah that a man once borrowed R. Landau's beautiful etrog in order to make a blessing on it on Sukkot. R. Landau then noticed the man reciting a לשם יחוד prior to his blessing. He snatched the etrog away, and said that anyone who says such prayers is not fit to make a blessing on his etrog!
5) Hakham Ovadia's son, R. David Yosef, began publishing the series Orhot Maran a few years ago, wherein he details several of the unique halakhic practices of his father. He related that Hakham Ovadia would generally omit the לשם יחוד prior to performance of prayers and missvot. There were a few notable exceptions, however, when he would say the לשם יחוד for a specific reason: a) Prior to counting the Omer - in order to state in the לשם יחוד that seffirat ha-Omer today is only rabbinically binding (a matter of dispute among the classic halakhic authorities), b) Prior to sitting in the Sukkah - because there is a prominent opinion in posekim that one must have in mind the reason for sitting in the Sukkah (זכר לענני הכבוד) in order to fulfill the missvah, c) Prior to blowing the shofar.