If your hand bleeds, but only when a napkin is applied, and one still has work to do, one
will need to wash one’s hand every now and then. Does that make the things one touches unclean? When one washes one’s hand, are the other parts of the hand also rendered unclean?
Or is it just that blood only comes out when a napkin is applied that makes only that part najis?

It is a contradiction to say:
a. Your hand is bleeding.
b. It only bleeds when covered by a napkin.
One has either to omit a. or b. from your question. It appears from what you have said that b. might be correct in your case. This implies that after you have washed your hands, all other parts of the kitchen are pure.