Praise be to Allah.
If there is a wound in any part of the body, then this wound is either uncovered or it is covered with a dressing or bandage.
If it is covered with a dressing or bandage, then he should wash the healthy parts of the body, then he should wet his hand with water and wipe over the dressing. If he does this then he does not need to do tayammum.
There are ahaadeeth which speak about wiping over splints, but they are all da’eef (weak), but that has been proven from ‘Abd-Allah ibn ‘Umar (may Allah be pleased with him).
Nothing has been proven from the Prophet (peace and blessings of Allah be upon him) on this topic… rather this is the view of the fuqaha’ among the Taabi’een and those who came after them, as well as what we have narrated from Ibn ‘Umar. He mentioned with his isnaad that Ibn ‘Umar (may Allah be pleased with him) did wudu when his hand was bandaged, and he wiped over it and over the dressing, and he washed everything else. He said: This is saheeh from Ibn ‘Umar."(Al-Majmoo’, 2/368).
But if the wound is uncovered, then it must be washed with water if possible, but if washing it will harm it, and he can wipe it then he should wipe it. If that is not possible, then the wound should be left without washing or wiping, then when the person has finished doing wudu, he should do tayammum.
Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) said in al-Sharh al-Mumti’ (1/169)
The scholars (may Allah have mercy on them) said: Wounds and the like are either uncovered or covered.
If the wound is uncovered then it must be washed with water. If it is not possible to wash it with water then it should be wiped over. If it is not possible to wipe over it then he should do tayammum. This is in order of preference.
If it is covered with some kind of appropriate material, then it can only be wiped. If wiping will harm it even though it is covered, then he should do tayammum, just as if it was uncovered. This is what was stated by the fuqaha’ (may Allah have mercy on them). End quote.
Shaykh Ibn Baz (may Allah have mercy on him) said:
If it has a dressing over it, he should wipe over it; if it is uncovered then he should do tayammum for it."(Fataawa Ibn Baz, 10/118)
Shaykh Saalih al-Fawzaan (may Allah preserve him) was asked:
After the washing that the doctor did for me, some blood came from my hand, from the site of the injection, so he put a bandage over it. If I take it off it starts to bleed and would not stop until night. This bandage is still wrapped around my left hand. Is it permissible for me wipe over it when I do wudu even though I was not taahir when it was put on, rather it was put on when there was bleeding; and how should I wipe over it?
You should not take off the bandage which is over your wound, especially since taking it off will harm you and make it bleed. It is not permissible for you to take it off in this case, because that poses a danger to you. So leave it where it is, and when you do wudu wash the part of the hand where there is no bandage, and for the part where the bandage is, it will be sufficient for you to wipe over the outside of it by wetting your hand with water and moving it over the outside of the bandage. This will suffice for you instead of washing what is beneath it for as long as it needs to stay, even if that is for many prayer times or for several days. It is not essential that the bandage have been put on when you were taahir, rather you may wipe over it, according to the correct view, even if you were not taahir at the time when it was put on and even if there is blood underneath it at the site of the injection or wound.
Conclusion: There is nothing wrong with you leaving the bandage on; rather you have to leave it on because it is in your interests. You can wipe over the outside of it when you wash the part of your hand that is not covered."(Al-Muntaqa min Fatawa al-Shaykh al-Fawzan, 5/15)
And Allah knows best.