Does Singer explore how the limits of one’s knowledge about the impacts of their actions might play into the decisions?

Only very briefly, and not in a way that I think really addresses your specific example:

Admittedly, it is possible that we are in a better position to judge what needs to be done to help a person near to us than one far away, and perhaps also to provide the assistance we judge to be necessary. If this were the case, it would be a reason for helping those near to us first. This may once have been a justification for being more concerned with the poor in one's town than with famine victims in India. Unfortunately for those who like to keep their moral responsibilities limited, instant communication and swift transportation have changed the situation. From the moral point of view, the development of the world into a "global village" has made an important, though still unrecognized, difference to our moral situation. Expert observers and supervisors, sent out by famine relief organizations or permanently stationed in famine-prone areas, can direct our aid to a refugee in Bengal almost as effectively as we could get it to someone in our own block. There would seem, therefore, to be no possible justification for discriminating on geographical grounds.