You have been already offered with two great dimension about your scenario being completely unrisky for HIV contraction, as stated by my fellow posters:
1. HIV can't infect once it is seperated from it's host.
2. Blood clotting would be an effective barrier to HIV transmission.
You must also understand another perspective:
3. We can't ignore the problem in estimating the risk of HIV transmission from breast milk. The real risk of transmission of HIV through breast milk is unknown. It is thought to be more likely if a mother is newly infected with "primary" infection and may therefore be more infectious. Women who are less viraemic are thought to be no to less likely to transmit HIV through breast milk.
Practically, the amount of viral content in breast milk might not be adequate to infect some one even if it establishes contact with a current gapping wound. Or, a very large quantity of breast milk would be necessary to form a contact with a wound that just occurred and so intense that it is bleeding profusely.
In totality, you should be more worried about dying this moment from falling off your bed while reading this response than dying due to an HIV related illness.
Breast milk is only a risk for infants with an underdeveloped immune system.
Secondly, handjobs, regardless of the presence of any cuts or scrapes or hangnails or anything else on a person's hands, are not a risk for HIV. It's literally the safest sexual contact you can possibly have, aside from complete abstinence. If the only type of sexual contact you ever have your whole life is handjobs, you will never get HIV.
Hi, no risk. I have stated before that a cut on one's finger clots very quickly. The point of clotting is to prevent any pathogen from entering the body. Long story short, you are at no risk for touching breast milk.