1. The Darlington circuit for relay coil drive looks strange. Why there is a resistor in collector? Normally you connect the coil here - two collectors short -> coil -> Vcc. Resistor between 1st emitter and 2nd base seems unnecessary.
1. Relay has a coil and contacts. Where is a second coil connection routed?
2. Schematics shows you short +5V with GND when relay is energised - is it really so?
3. Try to connect relay coil/Darlington GND and Vcc directly to the power source bypassing breadboard with possibly high AWG cable.
I personally use 4N33 for controlling relays. It gives an optoisolation so no return path for glitches through transistors.
HTH,
M.