Due to my awesome knife skills, I have a slightly deep cut on my thumb. I have three questions:
1: Do I have to take the bandage off when doing wudu to make sure that the water gets in contact with the skin?
2: If the cut keeps bleeding a bit even with the bandage on, does it nullify the wudu?
3: My cut isn’t too bad but if someone had an injury which they couldn’t put any water on, what are the guidelines of performing ablution in that case?
Re: Injury and Ablution
Peace Huma,
1 - As far I remember/read that there is a narration where we forbidden to undertake a difficult task which is beyond to our capability like clean the inside of the eyes while doing ablution. By keeping this in our mind, taking the bandage off when doing ablution is not a difficult task so one may do so.
2 - Some scholar said if the bleeding quantity is equal or more to the size of a coin then its nullify the ablution but some said regardless of the quantity of blood.
3 - 
Almighty Allah knows best.
Re: Injury and Ablution
If you think that its fairly easy for you to take bandage off and perform ablution, then do so.
If you know that injury is deep and contact with water may cause infection, you can do Tayammum.
Another thing thats my personal opinion is that you can do other steps of ablution, while avoiding water contact with wounded area and do Tayammum with it. It is our intention that counts most to Allah.
The concept of easiness is built into Islam and to not take advantage of the allowances that Allah has granted us is in fact na-shukri and sin. For example, it is permitted to shorten your prayers and skip fasting if you are on a journey.
NOTE: you do have to make up for the days of fasting that you skipped afterwards.