Connected Harbor Breeze light kit to ceiling fan. It is controlled by separate wall switches. I connected black to blue wire, and white to white. The wall switch for the light kit operates correctly. When I turn on the fan wall switch, the fan motor just hums. If I disconnect the light kit, the fan works fine.?

You say you added a light kit to an existing fan. You also say that the fan and the light are each controlled by a wall switch.
Did the two switches previously exist or did you add a switch to control the light?

How are the switches wired? How many and what color wires in switch box. How many and what color wires in ceiling box?