First of all, it is to be stated that the consensus of Muslim scholars agree that purity is a condition only in Tawaf (circumambulation of the Ka`bah), which means that a Muslim can perform all acts of Hajj or `Umrah, with the exception of Tawaf, without making wudu’ (ablution). However, Hanafi scholars and Imam Ibn Taymiyah consider that purity is not a condition even in Tawaf.
“As for the texts that require purity for making Tawaf, such as the Prophet’s words “A menstruating woman does all acts of Hajj save Tawaf” and also the hadith “When one of you gets impure, he should not pray unless after purification,” these texts denote general obligation which is to be observed according to one’s ability. Almighty Allah says: “So keep your duty to Allah as best ye can” (At-Taghabun: 16) The Prophet (peace and blessings be upon him) also said: “When I command you to do something, do it according to your capacity.” Attaining purity is known to be a condition pertaining to Tawaf. In case of Prayer, purity is more required and is considered to be a condition for Prayer being accepted. However, in cases of extreme necessity, this condition is neglected. This involves the case of a person with enuresis who can pray and make Tawaf after making ablution.”
So, having urine drops coming out of you at regular basis, you can act during the performance of Hajj and `Umrah exactly as you do for Prayer. You can apply a piece of cotton or a pad to the place of urinating as best as you can; you can put a rubber band over it, if you wish, so that no urine drops would fall and stain the Sacred Precinct.
Moreover, Sheikh M. S. Al-Munajjid, a prominent Saudi Islamic lecturer and author, states:
If you pass urine or drops of urine, this does not mean that you have to do ghusl (bathing); what you have to do is make wudu’ and wash only the parts of the body and clothing (on which the urine has fallen).
If the situation you describe — the occasional emission of a few drops of urine — is beyond your control, then your case is like that of one who is incontinent, so do the following:
1. Wash the private parts with water.
2. Wash the part of your clothes that the urine has gotten onto (there is no need to change your clothes).
3. Put a pad or a piece of cotton, or something similar, over the private parts so that the urine will not spread.
4. Do wudu’ for each Prayer, then pray as you are, and pray as much as you like, fard and nafl, with this wudu’. After doing this, anything else that comes out does not matter. Allah says: “So keep your duty to Allah and fear Him as much as you can.” (At-Taghabun: 16)
Do not miss any Prayers or delay doing them on time, and your Prayer will be valid.
This is what should be done in the case of incontinence. But if the urine comes out shortly after urinating then stops, in this case you can go to the bathroom fifteen minutes before the Prayer or the Adhan, for example, then put something (a cloth, for example) to prevent contamination after you do cleaning yourself after relieving yourself, perform wudu’ and pray.