There is nothing wrong with reciting the Qur’an while being in a state of minor impurity (without wudu.)
The following Hadith demonstrates that it is not necessary to do wudu if one wants to recite Qur’an and is in a state of minor impurity:
It was narrated from ‘Abd-Allah ibn ‘Abbaas (may Allah be pleased with him) that he stayed one night with Maymoonah, the Mother of the Believers, who was his maternal aunt. I lay down across the width of the mattress and the Messenger of Allah (blessings and peace of Allah be upon him) and his wife lay along its length. The Messenger of Allah (blessings and peace of Allah be upon him) slept until halfway through the night, or just before or after that. The Messenger of Allah (blessings and peace of Allah be upon him) woke up and started wiping the sleep from his face with his hand. Then he recited the last ten verses of Surat Aal ‘Imraan, then he got up and went to a waterskin that was hanging there, and did wudu from it and did it well, then he stood and prayed.
Narrated by al-Bukhaari, 4295; Muslim, 763
This Hadith indicates that Qur’an may be recited without wudu because he slept for a long time then he woke up and recited Qur’an before he did wudu, then he did wudu after that and prayed.
Rather the difference of scholarly opinion has to do with touching the Qur’an and whether one who is in a state of major impurity (janaabah) can recite it; the majority of scholars are of the view that both of these matters are not allowed because of the following Hadith:
‘Amr bin Hazm narrated that the Messenger of Allah (blessings and peace of Allah be upon him) said: “ None except a pure person should touch the Quran”.
[Reported by Malik as a Mursal and by An-Nasa’i and Ibn Hibban as Mawsul. And it is graded as Ma’lul (defective)].
As of reading Qur’an from your mobile phone while not being in a state of purity, mobile phones in which Qur’an is put in written or recorded form do not come under the same ruling as the Quran book, so it is permissible to touch them without having oneself being in a state of purity and it is permissible to take the device into the washroom with one because writing Qur’an on the mobile phone is not like writing it in the Qur’an book. The letters of the Quran in the app are not fixed, and the mobile phone may contain Qur’an and other things so the phone app cannot be defined as the Qur’an book. Based on this, it is permissible to touch the mobile phone while not being in a state of purity and it is permissible to read from it.
Reading Qur’an on a mobile phone makes it easy for women who are menstruating and one who cannot easily carry the Quran book with him, or one who is in a place where it is difficult for him to do wudu because it is not essential for the one who touches it to be in a state of purity, as mentioned above.