I am replacing a Trane mercury operated 4 wire thermometer (no heat pump) with
A Hunter Electronic Digital Thermostat Model 42999. Everything works except the
Blower will run on manual but not on automatic. The new thermometer has R, W, Y, and
G terminals. The RH and RC terminals are jumped.

If the blower will run in the on position but not in the auto when the cooling comes on you may have a thermostat that was bad out of the box. Some of the newer heating and air systems will use three different blower speeds on the control board, one for heating, one for cooling and one for just a blower call only. You may have a cooling blower relay going bad on the control board.

DITTO.

Buy a good quality thermostat and stay away from those HD cheapies.

Even Honeywell thermostats have some that are bad right out of the box.