Praise be to Allah.
This may be happening as the result of waswas (insinuating whispers) and may be something imaginary, which is from the Shaytan. Or it may really happen to some people. If it is really happening, he should not rush but wait until the urine has stopped, then he should wash his private part with water and that is the end of the matter. If he is worried that something may happen after that, he should sprinkle the parts of his clothes that are around the private part with water, then ignore anything that he may imagine after doing wudu, because this may help him to get rid of this waswas.
But if it is just his imagination or waswas, and is not real, then he should not pay any attention to it. The believer should not pay attention to such things, because this makes the Shaytan do it more. The Shaytan is keen to spoil the good deeds of the son of Adam, such as prayer and so on. So, it is essential to be aware of his tricks and whispers, and to rely on Allah and put one's trust in Him, and assume that what doubts he develops are from the Shaytan, so that he will not take any attention to that with regard to his wudu and then his prayer. He should not pay attention to any of these imaginary things. Then, if something really does come out of him, without a doubt, he should clean himself again and repeat his wudu.
But as for merely thinking it happened, he should not pay any attention to it; even if he thinks it is 90% possible that something has come out of him, he should not pay any attention to it, because this is something that will make the Shaytan do it to him again. So as long as he is not certain, he should carry on and pray and do wudu. Hence when the Prophet (blessings and peace of Allah be upon him) was asked: O Messenger of Allah, a man thinks that something came out of him whilst he was praying, he said: “He should not stop [praying] unless he hears a sound or notices a smell.” So the Prophet (blessings and peace of Allah be upon him) told them that he should not stop his prayer because of something he imagines, unless he hears a sound or notices a smell. So similarly, when a person has finished doing wudu, then he feels something, he should not go and do wudu again; rather he should continue with his wudu and prayer and other actions until he is 100% certain that something has come out. Otherwise, the basic principle is that nothing has come out of him and that this is waswas and a trick of the Shaytan, with which he tries to exhaust the believer and distract him by means of these things. We ask Allah to keep us safe and sound.
Shaykh ‘Abd al-‘Azeez ibn Baz (may Allah have mercy on him) .