Answer: No, the child cannot have a father with blood type A.
Explanation: The child whose blood type is B with a mother whose blood type is A can only have a father whose blood type is either B or AB. Each blood type is controlled by a pair of gene, a child inherits one from the father and one from the mother. For a child to have a blood type B when his mother has a blood type A, he must have inherited the B blood type from his father.
See the attached punnet squares for more information.