It is true of what your mum says. You should better not wash your face these days because the water may contain a lot of invisible bacterium that can intensify your pink eyes. You could just use the clean cloth to dip in the water and then squeeze it dry. Then you could use the clean cloth to rub your face. That could also be the good way.
Well, so sorry to know that you've got pink eyes. Anyway, it is true we have to take care of your eyes when there is pink eyes. But it is absolutely ok to wash your eyes. However, you gotta make sure you don't communicate the disease to other people, thus, you should keep your towel or basin well disinfected after use and make them exclusive. Just keep your face as what you do.
First of all, you had better not use makeup on the face in case of more serious infections, and not wearing makeup could make it easier to clean you face. Just use a clean towel to tenderly wipe your face, pay attention to avoiding touching the eyes. And after that, it is better to clean the towel in boiled water to kill the germs and sterilize the towel. In addition, you should not use commom water such as that from the tap to wash your eyes. This is because that there are bacteria in the water which might as well worsen your pink eyes. There are different kinds of medicine for pink eyes caused by different causes to wash eyes. For bacterial conjunctivitis, various antibiotic drops or ointments are available by prescription to speed up the healing process. For viral and allergic conjunctivitis, there are over-the-counter antihistamine or decongestant dropssuch as ocuhist, naphcon-A, and visine AC. You had better consult your doctor to see which kind of conjunctivitis taht your pink eyes belong to, and then use the correspondent medicine to wash the eyes, instead of using water. In addition, you should wipe clean the secreta in your eyes before using the drops or ointments.