بسم الله الرحمن الرحيم
In the above mentioned situation if you recite the holy Quran orally without touching it then there is no harm in it.
Yes, if you want to ask about reciting the Holy Quran by holding in hand then if you are mazur by Shariah then you may recite the Holy Quran with the wudhu which you make for salah. (If your wudhu breaks in between due to urine problem then there is no harm). In case you are not mazur by Shariah then it shall not be lawful for you to touch the Holy Quran without wudhu. Yes, if you hold the holy Quran with a cloth and turn its pages with a pen then you may recite in this way if your clothes turn impure very often. However, you should try your level best to maintain purity as per your capacity. In case you have some trouble you must give it up. Regarding the purity of the cloth some concessions are available with a patient specially for the recitation of the Holy Quran.
Mazur by Shariah is the person who suffers from the illness of urine drops or nose bleeding and the drops keep on coming continuously and he does not find a time in which he could offer salah with purity. Such a person is mazur. The ruling for him is that he should make wudhu for each salah and until the time of that salah remains his wudhu shall be intact. However if something other than your illness is experienced which breaks the wudhu then your wudhu shall break and you will have to make fresh wudhu.
For more details read the chapter “Mazur Ke Ahkam” in Bahishti Zewar, vol. 1, and apply it to your condition. Yet, if you have something to know then write to us.
As far as the matter of impurity of the cloth is concerned then if the cloth is smeared with urine and it is feared that it shall smear once again before completing the salah then it is not wajib to wash it, otherwise it is wajib to wash it. If it exceeds more than one rupee (width of a palm) then salah shall not be valid without washing it.
Allah knows Best!
Darul Uloom Deoband