When Mandan Mishra was defeated by Shankaracharya, then UbhayaBbharti asked questions about grihastha ashram and about science of sex. Then shankaracharya had to go into body of a king, to be able to answer those questions.
So exactly what were those questions asked by Bharti?
Ubhaya Bharati's questions are given in this excerpt from the Madhaviya Shankara Digvijaya, the traditional biography of Adi Shankaracharya:
Discuss with me the science and the art of love between the sexes (Kusumastra sastram). Enumerate its forms and expressions. What is its nature and what are its centres? How does it vary in the sexes during the bright and dark fortnights? What are its manifestations in man and woman?
Note that Adi Shankaracharya never actually answered these questions. This later excerpt from the Madhaviya Shankara Digvijaya says that when Adi Shankaracharya comes back for the debate, Ubhaya Bharati, who is considered by Advaitins to be an incarnation of Saraswati, leaves this world and goes to Brahmaloka.
But if for some reason you still want Adi Shankaracharya's insights on this subject, this excerpt from the Madhaviya Shankara Digvijaya relates that while Adi Shankaracharya inhabits the body of the king Amaruka, he composes a work on Kama:
Besides, he studies during this time the Sutras that the sage Vatsayana had made on this subject together with all the commentaries on the same, and had also produced a new work of great profundity on the theme, bearing the name of Amaruka.
That work is the Amaru Shatakam, and you can read it here. Some Advaitins claim that in this work Adi Shankaracharya merely used the language of Kama to convey the ideas of Advaita. Personally, I'm not a follower of Adi Shankaracharya so I'm skeptical of this story.
There have been several different digvijayas of Adi Sankara. Swami Tapasyananda goes into very lengthy scholarly discussion of the different ones as well as the history behind the one of the most accepted ones known as the Madhviya-sankara-vijaya in his Introduction to his translation of this work.
The discussion and points are lengthy. Since your question asks only about the 'supposed' argument between Sankara and Bharti, I will only address that as there are many problems and divergent views that are addressed in the Introduction. Swami Tapasyananda writes:
...The Editors of the 1868 edition, Navadweep Goswami and Jayanarayana Tarkapanchanana, have stated in their Preface that 'their edition has been prepared in the light of three texts they could get--one in Nagari letters which was procured with great difficulty, another in Telugu characters procured with equal difficulty; and still another in Bengali alphabets made on the basis of the above texts'. There is no reason why this text should not be given an equal importance as the one edited by Dr. Veezhinathan. According to the text of the Calcutta edition, Anantanandagiri is giving the history, not of 'Adi-Sankara who was born at Kaladi'. but of a Sankaracharya 'who was born immaculately to Visishta of Chidambaram', who continued to live at Chidambaram itself, took Sannyasa there, and who went on Dig-vijaya tours that are entirely different from the routes that Adi-Sankara us supposed to have taken in several of the other Vijayas. This Sankara is very largely concerned with reforming the various cults that prevailed in the country and very little with philosophy. The controversy with Mandana, which is one of the most glorious episodes in Adi-Sankara's life, finds a casual mention in the form of a synopsis. In this, as also in entering Amaruka's body and in the writing of the Bhashyas, the two Sankaracharyas are mixed up...So Anantanandagiri's text cannot be taken as a conclusive proof...It is only one of the traditions supported by some manuscripts. There is every possibility that this Chidambaram Sankaracharya is the Abhinava-Sankara, whom even modern scholars have mistakenly identified with Adi-Sankara and given 788 A.D. as his time. Besides, Anantanandagiri, the author, calls the hero of his work his Parama-guru (his teacher's teacher). This makes the matter all the more confusing. For, no one has recorded that Adi-Sankara or his disciples had a disciple called Anantanandagiri. Anandagiri (quite different from Anantanandagiri) was Sankara's disciple, and the Prachina-Sankara-vijaya attributed to him (a book quite different from Anantanandagiri's) is not available anywhere now. The point that we want to make out by these critical remarks is that it is not very desirable to take a dogmatic position on such points where no final view is possible with existing information. The best that can be said is that it is one of the the traditions.
So, as a poem, it is a unique and fitting tribute to Adi-Sankara. Whether one accepts all the mythological tales that occur are up to the reader.
