Praise be to Allaah.
If a person gets something impure on his body or clothes and he has wudoo’, then his wudoo’ is not broken by that, because none of the things that break wudoo’ have happened. But all that he has to do is to wash this impurity from his body or clothes, and pray with that wudoo’. There is nothing wrong with him doing that.
See al-Muntaqaa fi Fataawa Shaykh Saalih al-Fawzaan, part 3, p. 23
See also the answer to question # 5212.
When removing the impurity you have to do that with the left hand, not the right. If you need to immerse your hand in impure water for a reason – such as unblocking pipes or toilets – as mentioned in the question, do not use your hand directly, wear gloves or something similar. If you must touch the impurity then use your left hand, not your right.
And Allaah knows best.