What wll happen if the starting winding in a split pase motor fail to energize?

The motor would not start rotating. The motor will just hum and the rotor will be stationary. If you can get the rotor to start turning the motor will still run on its run winding.

I have a cement mixer that has a motor with a burnt out start winding. All I have to do is turn the barrel and the motor will start turning over. Without the start winding you have a choice of which way the motor will turn. If I turn the mixer barrel the opposite direction, the barrel will turn that way until I turn the unit off.