Praise be to Allah.
If the matter is as you describe, that a few drops of urine come out shortly after urinating, and you put a piece of cloth to prevent your body and clothing becoming contaminated, then when you want to do wudoo’ you have to remove the cloth and clean yourself, and wash the site where the urine has come out, or clean yourself with the handkerchief by wiping your private part, and it is essential that you wipe it no less than three times, because of the report narrated by Muslim (262) from Salmaan al-Farisi (may Allaah be pleased with him) who said: The Prophet (blessings and peace of Allah be upon him) forbade us to clean ourselves with less than three stones.
It is not permissible for the person who prays to carry any impurity during the prayer, and his prayer is not valid if he does so, according to the majority of scholars. See: al-Mughni (1/403) and al-Majmoo’ (3/157).
Based on that, if you carry the handkerchief in your pocket, your prayer is not valid. You should take out that handkerchief and place it beside you, or in your shoe that you have taken off, or you should keep with you a bag in which you may place the used handkerchief. If it so happened that you forgot the handkerchief in your pocket and you finished your prayer, then you do not have to do anything and your prayer is valid. If you remember it during the prayer, then you could take it out from your pocket and place it beside you, and be careful not to annoy anyone else with it.
It says in Nihaayat al-Muhtaaj: If it is the matter of a choice between throwing the impurity away now so that your prayer will be valid, but that requires throwing it in the mosque because you are there, or not throwing it away so as to protect the mosque from impurities but that will invalidate the prayer, then what I think is more appropriate is to pay attention to the validity of the prayer, and to throw the impurity away now in the mosque, then remove it immediately after finishing the prayer, because by doing that you can combine validity of the prayer with keeping the mosque clean. But it is acceptable to throw away the impurity and delay removing it until you have finished the prayer, because of the situation. End quote.
We do not think that keeping enough handkerchiefs to suffice you during the time you have to stay in the camp is something that will be too difficult for you, if you work out how many of them you need.
And Allah knows best.