Answer
Lymphocytes in this child's immune system would not be able to mature, produce antibodies, or identify and kill infected cells.
Work Step by Step
T cells mature in the thymus, and without this gland, they would not be able to function fully. Since mature T cells are needed to activate B cells, the lack of functional T cells also creates a lack of functional B cells. This in turn prevents the production of useful antibodies. The lack of mature T and B cells also prevents the immune system from identifying cells that have already been infected and destroying them. In short, this child's immune system would be largely non-functional, making the child extremely susceptible to infection and illness.