Answer
In the Name of Allah, Most Gracious, Most Merciful.
All praise and thanks are due to Allah, and peace and blessings be upon His Messenger.
In this fatwa:
There is nothing confirmed from the Prophet (peace be upon him) regarding Du`a during Wudu save the hadith of Abu Musa Al-Ash`ari.
In his famous work, Fiqh As-Sunnah, Sheikh Sayyid Sabiq, elaborates on the subject as follows:
There is nothing confirmed from the Prophet (peace be upon him) regarding Du`a during Wudu save the hadith of Abu Musa Al-Ash`ari who said, “I came to the Messenger of Allah with water. While he was performing Wudu, I heard him supplicate, ‘O Allah, forgive my sins. Make my residence spacious for me and bless me in my provisions.’ I said, ‘O Prophet of Allah, I heard you supplicating such and such.’ He said, ‘Did I leave anything out?” (Related by An-Nasa’i and Ibn As-Sunni with an authentic chain of transmission)
An-Nawawi includes this event under the chapter, What is to be said after one completes the Wudu, and Ibn as-Sunni has it under, What is to be said when one is in the state of ablution. An-Nawawi holds that both meanings may be implied from the Hadith.
Almighty Allah knows best.