The father and the boy have entered a house they assumed was empty. However, as they descended the cellar, looking for food, they came across a horrendous sight when they discovered humans who were being kept down there in shocking conditions. Clearly somebody was using the house as a store for humans which they were then feeding upon. As the boy and his father leave, they see people coming towards the house, who are obviously the people who live there and the people who have trapped the humans in the cellar for their own use. What concerns the father immediately about his cough therefore, as he and his son try to hide in a field, is that his cough will betray them and lead to their capture:
He was going to start coughing. He'd have to put his hand over his mouth but the boy was holding on to it and would not let go and in the other hand he was holding the pistol. He had to concentrate to stifle the cough and at the same time was trying to listen.
Initially therefore the cough concerns the father because he fears it will lead to the other humans finding them and their own capture. He is desperate to avoid this, partly because he has vowed that he would kill his son himself rather than let him be killed and eaten by others. However, in the long term, his cough concerns him because he fears it will lead to his declining health. As he is the only person who is looking after his son, he is greatly concerned about what might happen to his boy if he is not around any longer to look after him.