Injury, also known as physical trauma, is damage to the body caused by external force. This may be caused by accidents, falls, hits, weapons, and other causes. Major trauma is injury that has the potential to cause prolonged disability or death.
Thank you for your question. It is not considered tahir, but if the amount is less then the tip of the thumb to the first crease on your thumb, then it is alright to pray with that amount of blood as long as it has not come into contact with water. For the specific amount of blood there may be differences in opinion and so please also refer to the book of Islamic laws of your Marja.
May you always be successful
You would have to perform the ghusl in the the form of "ghusl jabirah", in the case where you are not allowed to get the injured hand wet.
Please watch a tutorial I have on this:
And Allah knows best.